NCERT Solutions for Class 12 Maths Chapter 3 Matrices

NCERT Maths Class 12 Chapter 3 Solutions, "Matrices," is based on the concept of Matrix and their properties. The chapter consists of the following exercises:

Exercise 3.1: This exercise introduces types of matrices and covers basic operations like addition, subtraction, and multiplication.

Exercise 3.2: This exercise explores matrix properties like commutative, associative, and identity matrix use.

Exercise 3.3: This exercise focuses on matrix multiplication, its properties, and practicing inverse calculations.

Exercise 3.4: This exercise teaches and applies the concept of matrix transpose and its properties.

Miscellaneous Exercise: Mix of questions testing all matrix concepts and problem-solving skills.

Overall, this chapter is an important topic in linear algebra and covers the fundamental concepts of matrices, including matrix notation, matrix operations, matrix properties, and matrix multiplication.

Some Important Points to remember

1. A matrix is said to be an ordered rectangular array of numbers or functions. These numbers or functions in the array are called the elements or the entries of the matrix.

2. Order of a Matrix

The order of the matrix determines the dimension of the matrix and the number of rows and columns in the matrix. The general representation of matrix order is Amxn, where m is the number of rows and n is the number of columns in the matrix.

3. Types of Matrices

• Column Matrix: A column matrix is a matrix with only one column.

• Row Matrix: A row matrix is a matrix with only one row.

• Square Matrix: A square matrix is one that has an equal number of rows and columns.

• Diagonal Matrix: A square matrix where only the diagonal elements are non-zero and all other elements are zero.

• Scalar Matrix: A diagonal matrix where all the diagonal elements are the same non-zero number.

• Identity Matrix: A diagonal matrix where all diagonal elements are '1' and all other elements are zero, often symbolized as 'I'.

• Zero or Null Matrix: A matrix where all elements are zero.

4. Equality of Matrices: Let A and B be two matrices. These matrices will be equal, if

(i) orders of A and B will be the same

(ii) corresponding elements of two matrices are the same

5. Operations on Matrices

• Addition of Matrices: You can add two matrices by adding the numbers that are in the same position in each matrix, as long as both matrices are the same size.

• Subtraction of Matrices: Subtracting matrices works like addition; you subtract the numbers in the same positions, but only if the matrices are the same size.

• Multiplication of Matrices: To multiply matrices, the number of columns in the first matrix must match the number of rows in the second; you then multiply and sum specific pairs of numbers.

6. Properties of Multiplication of Matrices

• Non-commutativity: 

Matrix multiplication will be not commutative i.e. if AB & BA are both defined, then it is not mandatory that AB ≠ BA.

• Associative law:

For three matrices A, B, and C, if multiplication is defined, then we can write it as A (BC) = (AB) C.

• Multiplicative identity: 

For any square matrix A, there will be an identity matrix of the same order in which  IA = AI = A.

Summary of NCERT Solutions Class 12 Matrices 

• A matrix is an ordered rectangular array of numbers or functions.

• A matrix having $m$ rows and $n$ columns is called a matrix of order $m \times n$.

• $\left[a_y\right]_{m \times 1}$ is a column matrix.

• $\left[a_h\right]_{\mathrm{b} x}$ is a row matrix.

• An $m \times n$ matrix is a square matrix if $m=n$.

• $A=\left[a_{i j}\right]_{\text {nex }}$ is a diagonal matrix if $a_{i j}=0$, when $i \neq j$.

• $A_y=\left[a_{i j}\right]_{n \times n}$ is a scalar matrix if $a_{i j}=0$, when $i \neq j, a_{i j}=k,(k$ is some constant $)$, when $i=j$.

• $A=\left[a_y\right]_{m \times n}$ is an identity matrix, if $a_{i j}=1$. when $i=j, a_{i j}=0$, when $i \neq j$.

• A zero matrix has all its elements as zero.

• $A=\left[a_i\right]-\left[b_y\right]-B$ if

1. A and B are of asme order,

2. $a_{i j}-b_{i j}$ for all possible values of $i$ and $j$.

• $k a=k\left[a_y\right]_{m \times n}-\left[k\left(a_y\right)\right]_{m \times n}$

• $-A=(-1) A$

• $A-B=A+(-1) B$

• $A+B=B+A$

• $(A+B)+C=A+(B+C)$, where $A, B$ and $C$ are of aame order.

• $k(A+B)=k A+k B$, where $\mathrm{A}$ and $\mathrm{B}$ of asme order, $k$ is constant.

• $(k+l) A=k A+l A$, where $k$ and $l$ are constant.

• If $A=\left[a_{j j}\right]_{m \beta n}$ and $B=\left[b_{i j}\right]_{N \times p}$, then $A B=C-\left[C_{j k}\right]_{w \times p}$, where $c_{j k}=\sum_{j=1}^n a_j b_{j k}$

1. $A(B C)=(A B) C$

2. $A(B+C)=A B+A C$

3. $(A+B) C=A C+B C$

•  If $A=\left[a_W\right]_{n \times N}$, then $A$ or $A^T=\left[a_N\right]_{N \times m}$

1. $\left(A^{\prime}\right)^{\prime}=A$,

2. $(k t)^{\prime}=k A^{\prime}$,

3. $(A+B)^{\prime}=A^{\prime}+B^{\prime}$;

4. $(A B)^{\prime}=B \cdot A^{\prime}$

• $A$ is a symmetric matrix if $A^{\prime}=A$.

• A is a skew symmetric matrix if $A^{\prime}=-A^{\prime}$.

• Any aquare matrix can be represented as the sum of a symmetric and a skew symmetric matrix.

• Elementary operations of a matrix are as follows:

1. $R_i \leftrightarrow R_j$ or $C_i \leftrightarrow C_i$

2. $R_l \rightarrow k R_i$ or $C_l \rightarrow k C_i$

3. $R_i \rightarrow R_i+k R_j$ or $C_i \rightarrow C_i+k C_j$

• If $A$ and $B$ re two square matrices such that $A B-B A=I$, then $B$ is the inverse matrix of $A$ and is denoted by $A^{-1}$ and $A$ is the inverse of $\mathrm{B}$.

Overview of Deleted Syllabus for Class 12 Maths Chapter 3

Class 12 Maths Chapter 3: Exercise Breakdown

Other Study Materials of CBSE Class 12 Maths Chapter 3

NCERT Solutions for Class 12 Maths | Chapter-wise List

Given below are the chapter-wise NCERT 12 Maths solutions PDF. Using these chapter-wise class 12th maths ncert solutions, you can get clear understanding of the concepts from all chapters.

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