medium difficulty9 min

Invertible Matrix and Inverse by Elementary Operations

A square matrix A is invertible if there exists a matrix A⁻¹ such that AA⁻¹ = A⁻¹A = I. A square matrix is invertible only when its determinant is non-zero.

In one line: Inverse turns A into identity; row operations must carry identity along.

Practice This Concept

Understand it

The inverse can be found using elementary row or column operations. For row operations, write A = IA and apply the same row operations to A and I until the left side becomes I; the right side then becomes A⁻¹. Row and column operations should not be mixed in one method.

Example

For A = [[1, 2], [3, 7]], determinant = 1 × 7 - 2 × 3 = 1, so A is invertible.

Formula and rule

Invertibility condition: A must be square and |A| ≠ 0.

For elementary row operations, transform [A | I] into [I | A⁻¹].

For 2 × 2 matrix [[a, b], [c, d]], inverse exists when ad - bc ≠ 0 and A⁻¹ = 1/(ad - bc) [[d, -b], [-c, a]].

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Find the inverse of A = [[1, 2], [3, 7]] by row operations.

1Write [A | I] = [[1, 2 | 1, 0], [3, 7 | 0, 1]].
2Apply R2 → R2 - 3R1: [[1, 2 | 1, 0], [0, 1 | -3, 1]].
3Apply R1 → R1 - 2R2: [[1, 0 | 7, -2], [0, 1 | -3, 1]].
4Hence A⁻¹ = [[7, -2], [-3, 1]].
5Check: determinant = 1, so the inverse is valid.

Method you can reuse

1Check that A is square.
2Compute determinant if practical.
3If determinant is zero, state inverse does not exist.
4If using row operations, form [A | I].
5Use only row operations to convert A into I.
6Read the transformed right side as A⁻¹.
7Verify by multiplying AA⁻¹ if needed.

Visual tip

No geometric diagram is required. The augmented matrix layout [A | I] is the useful visual form for row-operation questions.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Is A = [[2, 4], [1, 2]] invertible?

Think first, then show answer

No. Its determinant is 2 × 2 - 4 × 1 = 0, so A is not invertible.

Common wrong answer

Dividing by determinant zero in the 2 × 2 inverse formula, which falsely creates an inverse for a singular matrix.

Don't mix this up

A frequent error is mixing row and column operations while finding the inverse. This breaks the elementary operation method.

Exam tip

Start by checking determinant for a 2 × 2 or 3 × 3 square matrix. If determinant is zero, inverse does not exist and row-reduction work is unnecessary.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Asked as finding inverse using elementary row or column operations, checking invertibility, solving matrix equations, or using inverse properties in short and long answers.

1
Enough for 1 mark
A square matrix A is invertible if there exists a matrix A⁻¹ such that AA⁻¹ = A⁻¹A = I. A square matrix is invertible only when its determinant is non-zero.
2
Add for 2 marks
Example: For A = [[1, 2], [3, 7]], determinant = 1 × 7 - 2 × 3 = 1, so A is invertible.
3
Add for 3 marks
The inverse can be found using elementary row or column operations.

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