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

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 $1\times b$ 

2. 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 $a\times 1$ 

3. 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

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

5. 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.

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

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

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

$C=A+B$ 

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

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

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

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

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

3. 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 

$-A=\left( -1 \right)A$ 

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

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

4. 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.,

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

4. 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. $\left( A+B \right)C=AC+BC$ 

4. 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 ${{C}_{i}}\leftrightarrow {{C}_{j}}$.

2. 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

3. 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

• Matrices – Types, Operations, Properties, and Examples

• CBSE Class 12 Maths Chapter-3 Matrices Formula

• NCERT Solutions Class 12 Maths Chapter 3 Matrices

• Important Questions for CBSE Class 12 Maths Chapter 3 - Matrices