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Matrices Class 12 Notes CBSE Maths Chapter 3 (Free PDF Download)

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Revision Notes for CBSE Class 12 Maths Chapter 3 (Matrices) - Free PDF Download

Students need to manage various things during their academic life; hence it becomes imperative to seek professional help to get good grades. Math is a subject that can be used if concepts are cleared at the base level. That is what the scholars at Vedantu aim to do with their Matrices Class 12 Notes. Students will find the Class 12 Maths ch 3 Notes in-line with the latest CBSE curriculum and can be sure to score high with this material in their hands.

CBSE Class 12 Maths Revision Notes 2023-24 - Chapter Wise PDF Notes for Free

In the table below we have provided the PDF links of all the chapters of CBSE Class 12 Maths whereby the students are required to revise the chapters by downloading the PDF. 


Matrices Related Important Links

It is a curated compilation of relevant online resources that complement and expand upon the content covered in a specific chapter. Explore these links to access additional readings, explanatory videos, practice exercises, and other valuable materials that enhance your understanding of the chapter's subject matter.

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Find a curated selection of study resources for Class 12 subjects, helping students prepare effectively and excel in their academic pursuits.

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Matrices Class 12 Notes Maths - Basic Subjective Questions

Section–A (1 Mark Questions)

1. If $A=\left[\begin{array}{lll}2 & 1 & 3 \\ 4 & 5 & 1\end{array}\right]$ and,  then check whether $A B$ and $B A$ are defined or not.

Ans. Let $A=\left[a_{i j}\right]_{2 \times 3}$ and $B=\left[b_{i j}\right]_{3 \times 2}$. Both $\mathrm{AB}$ and $\mathrm{BA}$ are defined.


2. Identify the type of matrix $A=\left[\begin{array}{lll}0 & 0 & 5 \\0 & 5 & 0 \\5 & 0 & 0\end{array}\right]$

Ans. We know that, in a square matrix, the number of rows is equal to the number of columns. Hence, it is a square matrix.


3. If $\mathrm{A}$ and $\mathrm{B}$ are two matrices of the order $3 \times m$ and $3 \times n$, respectively and $\mathrm{m}=\mathrm{n}$, then find the order of matrix $(5 \mathrm{~A}-2 \mathrm{~B})$.

Ans. $A_{3 \times m}$ and $B_{3 \times n}$ are two matrices. If $m=n$, then $\mathrm{A}$ and $\mathrm{B}$ have same order as $3 \times n$ each, so the order of $(5 \mathrm{~A}-2 \mathrm{~B})$ should be same as $3 \times n$.


4. If $A=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right], B=\left[\begin{array}{lll}1 & 3 & 2 \\ 4 & 3 & 1\end{array}\right], C=\left[\frac{1}{2}\right], D=\left[\begin{array}{lll}4 & 6 & 8 \\ 5 & 7 & 9\end{array}\right]$, then which of the sums $\mathrm{A}+\mathrm{B}, \mathrm{B}+\mathrm{C}, \mathrm{C}+\mathrm{D}$ and $\mathrm{B}$ $+\mathrm{D}$ is defined?

Ans. Only $B+D$ is defined since matrices of the same order can only be added.


5. If $A=\left[\begin{array}{ll}5 & x \\ y & 0\end{array}\right]$ and $A=A^{\prime}$ then find the relation between $x$ and $y$.

Ans. $\because A=A^{\prime}$

$\begin{aligned} & \Rightarrow\left[\begin{array}{ll} 5 & x \\y & 0 \end{array}\right]=\left[\begin{array}{ll}5 & y \\x & 0\end{array}\right] \\& \Rightarrow x=y\end{aligned}$


Section–B (2 Mark Questions)

6. On using elementary column operations C_{2}\rightarrow C_{2}-2C_{1} in the following matrix equation

$$\left[\begin{array}{rr}1 & -3 \\2 & 4\end{array}\right]=\left[\begin{array}{rr}1 & -1 \\0 & 1\end{array}\right]\left[\begin{array}{ll}3 & 1 \\2 & 4\end{array}\right]$$ what do we get?

Ans. Given that, $\left[\begin{array}{rr}1 & -3 \\ 2 & 4\end{array}\right]=\left[\begin{array}{rr}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}3 & 1 \\ 2 & 4\end{array}\right]$

On using $C_2 \rightarrow C_2-2 C_1$


$$\left[\begin{array}{rr}1 & -5 \\2 & 0\end{array}\right]=\left[\begin{array}{rr}1 & -1 \\0 & 1\end{array}\right]\left[\begin{array}{rr}3 & -5 \\2 & 0\end{array}\right]$$


Since, on using elementary column operation on $X=A B$, we apply these operations simultaneously on $X$ and on the second matrix $B$ of the product $A B$ on $R H$.


7. Prove that Sum of two skew-symmetric matrices is always skew symmetric matrix.

Ans. Let $A$ is a given matrix, then (-A) is a skew-symmetric matrix. Similarly, for a given matrix $B$, (-B) is a skew-symmetric matrix. Hence, $-A-B=-(A+B)$ Sum of two skew-symmetric matrices is always skew - symmetric matrix.


8. If $A$ is a symmetric matrix, then prove that $A^3$ is a symmetric matrix.

Ans. If $A$ is a symmetric matrix, then $A^3$ is a symmetric matrix.

$\because A^{\prime}=A$

$$\begin{aligned}\therefore\left(A^3\right)^{\prime} & =A^3\quad\left[0\left(A^{\prime}\right)^n=\left(A^n\right)^1\right] \\&=A^3\end{aligned}$$


9. Construct A matrix $A=\left[a_{i j}\right]_{2 \times 2}$ whose elements ${ }^{a_{i j}}$ are given by $a_{i j}=e^{2 i x} \sin j x$

Ans. For $i=1, j=1, a_{11}=e^{2 x} \sin x$


For $i=1, j=2, a_{12}=e^{2 x} \sin 2 x$


For $i=2, j=1, a_{21}=e^{4 x} \sin x$


For $i=2, j=2 \cdot a_{22}=e^{4 x} \sin 2 x$


$$A=\left[\begin{array}{ll}e^{2 x} \sin x & e^{2 x} \sin 2 x \\e^{4 x} \sin x & e^{4 x} \sin 2x\end{array}\right]$$


10. If $A=\left[\begin{array}{ll}0 & a \\ 0 & 0\end{array}\right]$, then find the value of $A^{\text {in }}$

Ans. $A^2=\left[\begin{array}{ll}0 & a \\ 0 & 0\end{array}\right]\left[\begin{array}{ll}0 & a \\ 0 & 0\end{array}\right]=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$


$\therefore A^{16}=\left(A^2\right)^8=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]=O$


11. If $\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]\left[\begin{array}{ll}3 & 1 \\ 2 & 5\end{array}\right]=\left[\begin{array}{ll}7 & 11 \\ k & 23\end{array}\right]$, then find the value of $\mathrm{k}$.

Ans. 

$\left[\begin{array}{ll}1 & 2 \\3 & 4\end{array}\right]\left[\begin{array}{ll} 3 & 1 \\2 & 5\end{array}\right]=\left[\begin{array}{ll}7 & 11 \\k & 23\end{array}\right]$


$\Rightarrow \begin{bmatrix} 3+4 & 1+10 \\ 9+8& 3+20 \\ \end{bmatrix}=\begin{bmatrix} 7 & 11 \\ k & 23 \\ \end{bmatrix}$


$\Rightarrow \begin{bmatrix} 7 & 11 \\ 17 & 23 \\ \end{bmatrix}=\begin{bmatrix} 7 & 11 \\ k & 23 \\ \end{bmatrix}$


$\therefore k=17$



12. Find values of $a$ and $b$ if $A=B$, where $$A=\left[\begin{array}{cc}a+4 & 3 b \\8 & -6\end{array}\right] \text { and } B=\left[\begin{array}{cc}2 a+2 & b^2+2 \\8 & b^2-5 b\end{array}\right]$$


Ans. We have, $$A=\left[\begin{array}{cc}a+4 & 3 b \\8 & -6\end{array}\right]_{2 \times2} \text { and } B=\left[\begin{array}{cc}2 a+2 & b^2+2 \\8 & b^2-5 b\end{array}\right]_{2 \times 2}$$


Also, A = B, By equality of matrices we know that each element of A is equal to the corresponding element of B, 

$a_{11}= b_{11}\Rightarrow a+4=2a+2\Rightarrow a=2$

$a_{12}= b_{12}\Rightarrow 3b=b^{2}+2\Rightarrow b^{2}=3b-2$

And $a_{22}=b_{22}\Rightarrow -6=b^{2}-5b$

$\Rightarrow -6=3b-2-5b [\because b^{2}=3b-2]$

$\Rightarrow 2b=4\Rightarrow b=2$

$\therefore a=2\;and\;b=2$


13. Find non-zero values of $x$ satisfying the matrix equation: $$x\left[\begin{array}{cc}2 x & 2 \\3 & x\end{array}\right]+2\left[\begin{array}{ll}8 & 5 x \\4 & 4 x\end{array}\right]=2\left[\begin{array}{cc}\left(x^2+8\right) & 24 \\(10) & 6 x\end{array}\right]$$

Ans. Given that,$$x\left[\begin{array}{cc}2 x & 2 \\3 & x\end{array}\right]+2\left[\begin{array}{ll}8 & 5 x \\4 & 4 x\end{array}\right]=2\left[\begin{array}{cc}\left(x^2+8\right) & 24 \\(10) & 6 x \end{array}\right]$$

$$\begin{aligned}& \Rightarrow\left[\begin{array}{cc}2 x^2+16 & 2 x+10 x \\3 x+8 & x^2+8 x\end{array}\right]=\left[\begin{array}{cc}2 x^2+16 & 48 \\20 & 12 x \end{array}\right] \\& \Rightarrow 2 x+10 x=48 \\& \Rightarrow 12 x=48 \\& \therefore x=\frac{48}{12}=4\end{aligned}$$



PDF Summary - Class 12 Maths Matrices Notes (Chapter 3)

Matrix: 

  • It is an ordered rectangular array of collection of numbers or functions arranged in rows and columns is called matrix

  • The numbers or functions are known as the elements or entries of the matrix.

E.g. \[\left[ \begin{matrix} x & y  \\ 1 & 2  \\ \end{matrix} \right]\] 


Row and Column of a Matrix: 

  • The horizontal arrangement of elements or entries are said to form the row of a matrix 

  • The vertical arrangement of elements or entries are said to form the Column of a matrix. 

E.g.  \[\left[ \begin{matrix} x & y  \\ 1 & 2  \\ \end{matrix} \right]\] , This matrix has two rows and two columns.


Order of Matrix: 

  • It tells us about the number of rows and columns of a matrix.

  • It is represented by $a\times b$ means a matrix has $a$ rows and $b$ columns. 

  • For example: 

\[A=\left[ \begin{matrix} 2 & 8 & 3  \\ 1 & 9 & 8  \\ 0 & 7 & 0  \\ \end{matrix} \right]\], there are $3$ rows and $3$ columns therefore the order of matrix $A$ is $3\times 3$ 


Types of Matrices

  1. Row Matrix: A matrix containing only one row is known as row matrix.

  • For E.g.  $\left[ \begin{matrix} a  \\ b  \\ c  \\ \end{matrix} \right]$ 

  • The order of row matrix is $1\times b$ 

  1. Column Matrix: A matrix containing only one column is known as column matrix.

For E.g.  \[\left[ \begin{matrix} 1 & 2 & 3 & -2  \\ \end{matrix} \right]\] 

  • The order of column matrix is $a\times 1$ 

  1. Square Matrix: The number of rows and numbers of columns are equal in the matrix.

For E.g.  $\left[ \begin{matrix} 1 & 1 & 2  \\ 2 & 3 & 5  \\ 3 & 6 & 8  \\ \end{matrix} \right]$ 

  • The order of square matrix is always $a\times a$, where $a$ can be any natural number

  1. Diagonal Matrix: If the diagonal elements are non-zero and all the non-diagonal elements of a matrix are zero, then such type of matrix is known as Diagonal Matrix.

For E.g.  $\left[ \begin{matrix} 1 & 0 & 0  \\ 0 & 2 & 0  \\ 0 & 0 & 5  \\ \end{matrix} \right]$ 

  1. Scalar Matrix: It is a type of diagonal matrix in which all diagonal elements are equal.

For E.g.  $\left[ \begin{matrix} x & 0  \\ 0 & x  \\ \end{matrix} \right],\left[\begin{matrix} 4 & 0 & 0  \\ 0 & 4 & 0  \\ 0 & 0 & 4  \\ \end{matrix} \right]$ etc.

  1. Identity Matrix: It is a type of diagonal matrix in which all diagonal elements are equal to $1$.

For E.g.  $\left[ \begin{matrix} 1 & 0 & 0  \\ 0 & 1 & 0  \\ 0 & 0 & 1  \\ \end{matrix} \right]$ 

  1. Zero Matrix: In it all the elements are zero and this is also known as null matrix.

For E.g.  $\left[ \begin{matrix} 0 & 0  \\ 0 & 0  \\ \end{matrix} \right],\left[ \begin{matrix} 0 & 0 & 0  \\ \end{matrix} \right]$ etc.


Equality of Matrices: 

  • Two matrices are equal if and only if the order of both the matrices are equal and the element of one matrix is equal to the corresponding element of another matrix.

For E.g.  $A={{\left[ \begin{matrix} 1 & 8  \\ 8 & 4  \\ \end{matrix} \right]}_{2\times 2}}$  and  $B={{\left[ \begin{matrix} 1 & 8  \\ 8 & 4  \\ \end{matrix} \right]}_{2\times 2}}$ 

All the elements of matrix $A$ are equal to the corresponding elements of        matrix $B$ and the order of both matrices is the same. Hence, $A=B$. 


Operations in Matrices

  1. Addition of Matrices: 

  • Addition of two matrices can be done only when they have the same order.

  • Addition can be done by adding the corresponding entries of the two matrices

For e.g.  $A=\left[ \begin{matrix} 1 & 0  \\ 7 & 4  \\ \end{matrix} \right]\;and\;B=\left[ \begin{matrix} 2 & 1  \\ 3 & 5  \\ \end{matrix} \right]$ 

$C=A+B$ 

$C=\left[ \begin{matrix} 1 & 0  \\ 7 & 4  \\ \end{matrix} \right]+\left[ \begin{matrix} 2 & 1  \\ 3 & 5  \\ \end{matrix} \right]$ 

$C=\left[ \begin{matrix} 3 & 1  \\ 10 & 9  \\ \end{matrix} \right]$ 

  1. Multiplication of a Matrix by a scalar: 

  • When a matrix is multiplied by a scalar, then each element of the matrix is multiplied by the scalar quantity and a new matrix is obtained.

For E.g.  $2\left[ \begin{matrix} 4 & 5  \\ 6 & 7  \\ \end{matrix} \right]$ 

=$\left[ \begin{matrix} 4\times 2 & 5\times 2  \\ 6\times 2 & 7\times 2  \\ \end{matrix} \right]$ 

=$\left[ \begin{matrix} 8 & 10  \\ 12 & 14  \\ \end{matrix} \right]$ 

  1. Negative of a Matrix: 

  • Multiplying a matrix by $-1$ gives negative of that matrix

For E.g. $A=\left[ \begin{matrix} 1 & -1  \\ -1 & -2  \\ \end{matrix} \right]$ 

Negative of Matrix $A$ is 

$-A=\left( -1 \right)A$ 

$-A=\left( -1 \right)\left[ \begin{matrix} 1 & -1  \\ -1 & -2  \\ \end{matrix} \right]$ 

$-A=\left[ \begin{matrix} -1 & 1  \\ 1 & 2  \\ \end{matrix} \right]$ 

  1. Difference of Matrices:

  • Two matrices can be subtracted only when they have same order

  • Subtraction can be done by subtracting the corresponding entries of the two matrices

For e.g.  $A=\left[ \begin{matrix} 1 & 6  \\ 7 & 4  \\ \end{matrix} \right]\;and\;B=\left[ \begin{matrix} 2 & 1  \\ 7 & 9  \\ \end{matrix} \right]$

$C=A-B$ 

$C=\left[ \begin{matrix} 1 & 6  \\ 7 & 4  \\ \end{matrix} \right]-\left[ \begin{matrix} 2 & 1  \\ 7 & 9  \\ \end{matrix} \right]$ 

$C=\left[ \begin{matrix} -1 & 5  \\ 0 & -5  \\ \end{matrix} \right]$ 


Properties of Matrix Addition

  1. Commutative Law: Matrix addition is commutative i.e., $A+B=B+A$ . 

  2. Associative Law: Matrix addition is associative i.e.,$\left( A+B \right)+C=A+\left( B+C \right)$ . 

  3. Existence of Additive Identity: Zero matrix $O$ is the additive identity of a matrix because adding a matrix with zero matrix leaves it unchanged i.e.,

$X+O=O+X=X$ .

  1. Existence of Additive Inverse: Additive inverse of a matrix is a matrix which on adding with another matrix yield $0$ i.e., $X+\left( -X \right)=\left( -X \right)+X=0$ 


Multiplication of Matrices: 

  • Multiplication of two matrices $A$ and $B$ is defined when the number of columns of $A$ is equal to the number of rows of $B$. 

  • Entries in rows are multiplied by corresponding entries in columns i.e., entries in the first row are multiplied by entries in the first column and similarly for other entries. 

E.g. $A=\left[ \begin{matrix} 2 & 1  \\ 1 & 2  \\ \end{matrix} \right]$ and $B=\left[ \begin{matrix} 0 & 2 & 1  \\ 1 & 1 & 1  \\ \end{matrix} \right]$ 

Product of $A$ and $B$ is 

$AB=\left[ \begin{matrix} 2\left( 0 \right)+1\left( 1 \right) & 2\left( 2 \right)+1\left( 1 \right) & 2\left( 1 \right)+1\left( 1 \right)  \\ 1\left( 0 \right)+2\left( 1 \right) & 1\left( 2 \right)+2\left( 1 \right) & 1\left( 1 \right)+2\left( 1 \right)  \\ \end{matrix} \right]$  

$AB=\left[ \begin{matrix} 1 & 5 & 3  \\ 2 & 4 & 3  \\ \end{matrix} \right]$ 


Properties of Matrix Multiplication

  1. Non-Commutative Law: Matrix multiplication is not commutative i.e., $AB\ne BA$ but not in the case of a diagonal matrix.

  2. Associative Law: Matrix multiplication follow associative law i.e., $A\left( BC \right)=\left( AB \right)C$ 

  3. Distributive Law: Matrix multiplication follow distributive law i.e.,

  1. $A\left( B+C \right)=AB+AC$ 

  2. $\left( A+B \right)C=AC+BC$ 

  1. Existence of Multiplicative Identity: Identity matrix $I$ is the multiplicative identity of a matrix because multiplying a matrix with $I$ leaves it unchanged.


Transpose of a Matrix:

  • It is the matrix obtained by interchanging the rows and columns of the original matrix.

  • It is denoted by ${{P}^{'}}$ or ${{P}^{T}}$ if the original matrix is $P$.

For E.g.  $P=\left[ \begin{matrix} 1 & 2  \\ 3 & 4  \\ \end{matrix} \right]$ ${{P}^{T}}or{{P}^{'}}=\left[ \begin{matrix} 1 & 3  \\ 2 & 4  \\ \end{matrix} \right]$ 


Properties of Transpose of Matrix

  1. $\left( A' \right)'=A$ 

  2. $\left( kA \right)'=kA'$ (Where, $k$ is any constant)

  3. $\left( A+B \right)'=A'+B'$ 

  4. $\left( AB \right)'=B'A'$


Special Types of Matrices

  • Symmetric Matrices: It is a square matrix in which the original matrix is equal to its transpose.

For E.g.  $P=\left[ \begin{matrix} 1 & -1 & 3  \\ -1 & 2 & 7  \\ 3 & 7 & 3  \\ \end{matrix} \right]$  

Transpose of Matrix $P$, ${{P}^{T}}=\left[ \begin{matrix} 1 & -1 & 3  \\ -1 & 2 & 7  \\ 3 & 7 & 3  \\ \end{matrix} \right]$

$\because P={{P}^{T}}$ 

Therefore, it is a Symmetric Matrix.

  • Skew-Symmetric Matrices: It is a square matrix in which the original matrix is equal to the negative of its transpose.

For E.g. $P=\left[ \begin{matrix} 9 & 2 & -3  \\ -2 & 0 & 7  \\ 3 & -7 & 0  \\ \end{matrix} \right]$  

Transpose of Matrix $P$, \[{{P}^{T}}=\left( -1 \right)\left[ \begin{matrix} 9 & 2 & -3  \\ -2 & 0 & 7  \\ 3 & -7 & 0  \\ \end{matrix} \right]\]

$\because {{P}^{T}}=-P$ 

Therefore, it is a Skew-Symmetric Matrix.


Elementary Operation (Transformation) of a Matrix

Elementary operations can be performed by three ways

  1. By interchanging any two rows or two columns. 

  • Interchange of ${{i}^{th}}$ and ${{j}^{th}}$ rows is denoted as ${{R}_{i}}\leftrightarrow {{R}_{j}}$ 

  • Interchange of ${{i}^{th}}$ and ${{j}^{th}}$ columns is denoted by ${{C}_{i}}\leftrightarrow {{C}_{j}}$.

  1. By multiplying any scalar to each element of any row or column of matrix.

  • It is denoted as ${{R}_{i}}\leftrightarrow k{{R}_{j}}$ for rows and ${{C}_{i}}\leftrightarrow k{{C}_{j}}$ for columns

  1. By multiplying any scalar to each element of any row or column and then adding the result to any other row or column. 

It is denoted as ${{R}_{i}}\leftrightarrow {{R}_{i}}+k{{R}_{j}}$for rows and ${{C}_{i}}\leftrightarrow {{C}_{i}}+k{{C}_{j}}$ for column.

Invertible Matrix

  • A matrix $A$ is invertible only when there exists another matrix $B$ such that $AB=BA=I$ , where $I$ is an identity matrix.

  • It is a property of the square matrix.

  • Inverse of the matrix is always unique.

For E.g. – Let us consider two matrices 

$A=\left[ \begin{matrix} 2 & 3  \\ 2 & 2  \\ \end{matrix} \right]$  and 

$B=\left[ \begin{matrix} -1 & \dfrac{3}{2}  \\ 1 & -1  \\ \end{matrix} \right]$

Now, 

$AB=\left[ \begin{matrix} 2 & 3  \\ 2 & 2  \\ \end{matrix} \right]\left[ \begin{matrix} -1 & \dfrac{3}{2}  \\ 1 & -1  \\ \end{matrix} \right]$

$=\left[ \begin{matrix} 1 & 0  \\ 0 & 1  \\ \end{matrix} \right]$ 

$=I$ 

And 

$BA=\left[ \begin{matrix} -1 & \dfrac{3}{2}  \\ 1 & -1  \\ \end{matrix} \right]\left[ \begin{matrix} 2 & 3  \\ 2 & 2  \\ \end{matrix} \right]$ 

$=\left[ \begin{matrix} 1 & 0  \\ 0 & 1  \\ \end{matrix} \right]$ 

$=I$  

Hence, $B$ is inverse of $A$ 

Inverse of a Matrix by Elementary operations

  • Inverse of a matrix can be obtained by using elementary operations.

  • We know that $A=IA$ on using elementary operation on $A$only which is on the left side of  equal to keeping right side one as it is and on $I$ then the identity matrix $I$ will become inverse of $A$ 

For example: Inverse of  

$A=\left[ \begin{matrix} 3 & 2  \\ 1 & 4  \\ \end{matrix} \right]$  using elementary operation.

We know that $A=IA$ 

$\left[ \begin{matrix} 3 & 2  \\ 1 & 4  \\ \end{matrix} \right]=\left[ \begin{matrix} 1 & 0  \\ 0 & 1  \\ \end{matrix} \right]A$ 

${{R}_{1}}\to \dfrac{{{R}_{1}}}{3}$ 

$\left[ \begin{matrix} 1 & \dfrac{2}{3}  \\ 1 & 4  \\ \end{matrix} \right]=\left[ \begin{matrix} \dfrac{1}{3} & 0  \\ 0 & 1  \\ \end{matrix} \right]A$

${{R}_{2}}\to {{R}_{2}}-{{R}_{1}}$ 

$\left[ \begin{matrix} 1 & \dfrac{2}{3}  \\ 0 & \dfrac{10}{3}  \\ \end{matrix} \right]=\left[ \begin{matrix} \dfrac{1}{3} & 0  \\ \dfrac{-1}{3} & 1  \\ \end{matrix} \right]A$

${{R}_{2}}\to {{R}_{2}}\times \dfrac{3}{10}$ 

$\left[ \begin{matrix} 1 & \dfrac{2}{3}  \\ 0 & 1  \\ \end{matrix} \right]=\left[ \begin{matrix} \dfrac{1}{3} & 0  \\ \dfrac{-1}{10} & \dfrac{3}{10}  \\ \end{matrix} \right]A$

${{R}_{1}}\to {{R}_{1}}-\dfrac{2}{3}{{R}_{2}}$ 

$\left[ \begin{matrix} 1 & 0  \\ 0 & 1  \\ \end{matrix} \right]=\left[ \begin{matrix} \dfrac{2}{5} & \dfrac{-1}{5}  \\ \dfrac{-1}{10} & \dfrac{3}{10}  \\ \end{matrix} \right]A$

Since, $I={{A}^{-1}}A$ 

Therefore, ${{A}^{-1}}=\left[ \begin{matrix} \dfrac{2}{5} & \dfrac{-1}{5}  \\ \dfrac{-1}{10} & \dfrac{3}{10}  \\ \end{matrix} \right]$ (Not in the current syllabus)


Other Related Links for Class 12 Maths Chapter 3


Conclusion

Vedantu's Revision Notes for CBSE Class 12 Maths Chapter 3 on Matrices provide comprehensive and well-structured study material to enhance understanding and preparation for exams. The notes cover all the important concepts, formulas and key points related to matrices in a concise manner. The content is presented in a student-friendly language, making it easy to grasp complex topics. Additionally, the notes include solved examples and practice questions that help students improve their problem-solving skills. With Vedantu's Revision Notes, students can confidently approach the subject, reinforce their knowledge and achieve success in their Class 12 Maths examinations.

FAQs on Matrices Class 12 Notes CBSE Maths Chapter 3 (Free PDF Download)

1. What is the Transpose of a Matrix and its properties?

For a matrix X = [xij]mxn, we get its transpose by interchanging its rows with columns. The transpose matrix will be of order n x m and is denoted by a. So, the transpose of X is X’ or XT. We will clarify this with an example:


X = |1 2|

      |3 4|

      |5 6|  then transpose of X, XT = |1 3 5|

                                                               |2 4 6|


The properties of a transpose are:

  • (X’)’ = X

  • (X + Y)’ = X’ + Y’

  • (XY)’ = Y’X’

  • (kX)’ = kX’

  • (XYZ)’ = Z’Y’X’

2. What are Invertible Matrices and Their Properties?

A square matrix X of order m is said to be invertible if there exists another square matrix Y (of the same order m) such that XY = YX = 1, then Y is called the inverse matrix of X and is represented by X-1. Let us see an example of the inverse of a matrix:


X =|2 3|

     |1 2| 


and Y = |2 -3|

              |-1 2|


XY =  |4 - 3  -6 + 6|

          |2 - 2  -3 + 4|


|2 -3|

|-1 2|        = 1


Below are listed a few of the properties of invertible matrices:

  • If Y is the inverse of X, then X is also an inverse of Y.

  • To be invertible, matrices need to be square matrices and must have the same order. That is why rectangular matrices are not invertible.

3. What do you mean by matrices?

Matrices are rectangular arrays of integers with rows and columns. Matrices are used to execute mathematical operations such as addition, multiplication, subtraction, and division. The numbers are referred to as matrix elements or entries. Matrices find their extensive usage in the domains of engineering, physics, economics, and statistics, as well as in many fields of mathematics. The size of the matrix is referred to as its order, and it is represented by rows and columns. Rows are always stated first in practice.

4. What do you mean by a scalar matrix?

A scalar matrix is a square matrix with a constant value as every element of its major diagonal and zeros for all other elements. A scalar matrix has all off-diagonal members equal to 0, and the values exhibited by all on-diagonal elements is 1. It is defined as a multiple of an identity matrix with any scalar quantity. It is a square matrix that exhibits the same number of rows and columns. To know more about matrices, you can download the Vedantu revision notes PDF of Chapter 3 free of cost from the website and app.

5. How can we utilise matrices in our day-to-day life?

Matrices are used to create graphs and statistics, as well as to conduct scientific investigations and research in a variety of disciplines. Matrices may also be used to represent real-world statistics such as population, infant mortality rate, and so on. Matrices are frequently used in engineering, physical problems, statistics, and economics, as well as many other domains related to mathematics. Multiplying matrices can provide rapid but accurate approximations to considerably more difficult calculations in many time-critical engineering applications.

6. What are the different types of matrices in Chapter 3 of Class 12 Maths?

Row matrices, column matrices, null matrices, square matrices, diagonal matrices, upper triangular matrices, lower triangular matrices, symmetric matrices, and asymmetric matrices are some of the several forms of matrices. All off-diagonal elements in a scalar matrix are equal to zero, but all non-diagonal elements are equal. Proper knowledge of these types and the ways to solve problems related to them is very crucial as they carry a significant weightage in the board’s paper.

7. Is multiplication of matrices distributive in nature?

As with real numbers, matrices may be distributed similarly. Check that each product of a matrix A has A on the left if the matrix A is distributed from the left! Matrix product requires that the entries belong to a semiring, but it is not necessary that the multiplication of the semiring's elements be commutative. The product is not commutative in general for matrices over fields; however, it is associative and distributive over matrix addition.