Matrices 3.4 NCERT Solutions

Matrices: Chapter 3 Links

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This exercise of Matrices 3.4 of Class 12 Maths Chapter 3 focuses on the concept of invertibility of matrices. It will help you understand when a matrix has an inverse, how inverse matrices are defined and the important properties related to matrix inversion.

Moreover, we will study key conditions such as the existence of inverse only for square matrices, the uniqueness of inverse, and the fundamental identity condition $AB = BA = I$ . These results are essential for solving advanced topics like systems of linear equations in the next chapter on determinants.

We will also verify important theorems like the uniqueness of inverse and the inverse of product of matrices, which form the theoretical backbone of matrix operations in Class 12 Mathematics.

Key Concepts

Invertible Matrix: A square matrix is called invertible if there exists another matrix such that their product gives identity matrix.
Inverse Condition: A matrix A has an inverse B if $AB = BA = I$ , where I is the identity matrix. Remember, if A is inverse of B, then B is also inverse of A.
Square Matrix Requirement: Only square matrices (m × m) can have inverses. Since for products $AB$ and $BA$ to be defined and to be equal, it is necessary that they should be square matrices of the same order.
Identity Matrix: A matrix I is called identity matrix if it has 1s on the main diagonal and 0 elsewhere, and $AI = IA = A$ .
Uniqueness of Inverse: If a matrix has an inverse, then it is unique (only one inverse exists).
Inverse of Inverse: If $B = A^{-1}$ , then $A = B^{-1}$ .
Product of Inverses: If A and B are invertible matrices of same order, then $(AB)^{-1} = B^{-1}A^{-1}$ . (Reversal Law)
Order Importance: In matrix multiplication, order matters, so inverse of product reverses the order.
Verification Method: To check inverse, multiply matrices and verify if result is identity matrix.
Important Note: In this exercise, inverse is verified using multiplication method only (determinant/adjoint method is covered in later chapter).

Now, let us begin with some examples first.

Example 1: Matrices 3.4

1. If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} -2 & 1 \\ 1.5 & -0.5 \end{bmatrix}$ then verify that $AB = BA = I$ , hence show that A and B are inverses of each other.

Solution

First, find $AB$ .

$AB = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} -2 & 1 \\ 1.5 & -0.5 \end{bmatrix}$

$= \begin{bmatrix} (1\times -2 + 2\times 1.5) & (1\times 1 + 2\times -0.5) \\ (3\times -2 + 4\times 1.5) & (3\times 1 + 4\times -0.5) \end{bmatrix}$

$= \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I$

Now, find $BA$ .

$BA = \begin{bmatrix} -2 & 1 \\ 1.5 & -0.5 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = I$

Hence,

$AB = BA = I$

Therefore, A and B are inverses of each other.

Example 2: Inverse Matrix

2. If $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ then verify that $AB = BA = I$ , hence show that A and B are inverses of each other.

Solution

First, find $AB$ .

$AB = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

$= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = I$

Similarly,

$BA = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = I$

Hence,

$AB = BA = I$

Therefore, A and B are inverses of each other.

You may already be following Maths Better for important concepts and exam-oriented questions, ranging from integration tricks to solving linear equations using the Matrix Method. Similarly, this NCERT Solutions series for Class 12 Maths will help strengthen your concepts and improve problem-solving skills.

Now, let’s understand two important theorems before moving on to the only question of Matrices 3.4.

Theorem 1: Uniqueness of Inverse

Statement: The inverse of a square matrix, if it exists, is unique.

Proof:

Let A be a square matrix of order m.

Assume that A has two inverses B and C.

Then, by definition of inverse:

$AB = BA = I$

and

$AC = CA = I$

Now consider:

$B = BI$

Substitute I = AC:

$B = B(AC)$

Using associativity:

$B = (BA)C$

Since $BA = I$ , we get:

$B = IC = C$

Hence,

$B = C$

Therefore, the inverse of a matrix is unique.

Theorem 2: Inverse of Product of Matrices

Statement: If A and B are invertible matrices of the same order, then the inverse of their product is given by:

$(AB)^{-1} = B^{-1}A^{-1}$

Proof:

We start from the definition of inverse:

$(AB)(AB)^{-1} = I$

Now pre-multiply both sides by $A^{-1}$ :

$A^{-1}(AB)(AB)^{-1} = A^{-1}I$

Using associativity:

$(A^{-1}A)B(AB)^{-1} = A^{-1}$

Since $A^{-1}A = I and A^{-1}I = A^{-1}$ , we get:

$B(AB)^{-1} = A^{-1}$

Now pre-multiply both sides by $B^{-1}$ :

$B^{-1}B(AB)^{-1} = B^{-1}A^{-1}$

So,

$I(AB)^{-1} = B^{-1}A^{-1}$

Hence,

$(AB)^{-1} = B^{-1}A^{-1}$

✔ Hence proved.

Matrix-based questions are generally direct, but presentation and correct steps play an important role in scoring full marks.

Here’s the only question of Ex – 3.4.

Question 1: Exercise 3.4 Matrices

1. Matrices $A$ and $B$ will be inverse of each other only if

(A) $AB = BA$
(B) $AB = BA = O$
(C) $AB = O, BA = I$
(D) $AB = BA = I$

Solution

By definition of inverse of a matrix, two matrices A and B are inverses of each other if their product gives identity matrix.

So, the condition is:

$AB = BA = I$

Where I is the identity matrix.

This is the fundamental condition for invertibility.

✔ Therefore, the correct option is:

(D) $AB = BA = I$

NCERT Class 12 Maths has a total of 61 exercises across Part 1 and Part 2. I’ll be covering each exercise one by one with clear explanations and exam-oriented solutions. Many of these questions are also available in video format on my YouTube channel, @Mathsbetter.

Common Mistakes to Avoid

Confusing inverse condition: Many students wrongly assume that only $AB = I$ is enough. In reality, both conditions are required: $AB = BA = I$
Ignoring order of multiplication: Matrix multiplication is not commutative. So $AB \ne BA$ in general, but for inverses both must be equal to identity matrix.
Skipping verification: In exam questions, students sometimes directly assume matrices are inverses without verifying multiplication. In fact, this verification also acts as a useful tool to check whether the calculated inverse is correct or not.
Incorrect identity matrix: Some students make mistakes in writing identity matrix, especially in 3×3 cases where diagonal must be 1 and all other elements 0.
Sign and arithmetic errors: Small calculation mistakes during multiplication often lead to wrong conclusion about inverse.

Continue Learning

Now that you have completed Matrices 3.4, you should revise the following related concepts to strengthen your understanding of matrix operations and inverses: