If A = bmatrix 1 & -2 & 0 \\ 2 & -1 & -1 \\ 0 & -2 & 1 bmatrix, find A^-1 and use it to solve the following system of equations : x - 2y = 10, 2x - y - z = 8, -2y + z = 7

A^-1=bmatrix-3&2&2\\-2&1&1\\-4&2&3bmatrix; x = 0, y = −5, z = −3

MathsCBSE Class 12 · Board exam 2024, set 65/1/1 · 5 marks · Long answer

CBSE Class 12 Mathematics 2024: question 32(a)

Set in the CBSE Class 12 board exam of 2024, paper 65/1/1, section D, question 32(a), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32(a) · 5 marks · Long answerMatrices

If $A = 120 - 2 - 1 - 2 0 - 1 1$ , find $A^{- 1}$ and use it to solve the following system of equations :
$x - 2 y = 10$ , $2 x - y - z = 8$ , $- 2 y + z = 7$

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Start by finding |A|; if non-zero, the inverse exists.

Hint 2

Find all nine cofactors and transpose to get adj A.

Hint 3

The coefficient matrix of the system is A itself, so X = A⁻¹B.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

$∣ A ∣ = 1 [(- 1) (1) - (- 1) (- 2)] - (- 2) [2 \cdot 1 - (- 1) \cdot 0] + 0 = - 3 + 4 = 1 \neq = 0$ , so $A^{- 1}$ exists.
1 mark
Cofactors: $C_{11} = - 3, C_{12} = - 2, C_{13} = - 4; C_{21} = 2, C_{22} = 1, C_{23} = 2; C_{31} = 2, C_{32} = 1, C_{33} = 3$ . So $adj A = - 3 - 2 - 4 212213$
2 marks
$A^{- 1} = \frac{1}{∣ A ∣} adj A = - 3 - 2 - 4 212213$
½ mark
The system is $A X = B$ with $X = x y z$ , $B = 1087$ . So $X = A^{- 1} B = - 30 + 16 + 14 - 20 + 8 + 7 - 40 + 16 + 21 = 0 - 5 - 3$ , i.e. $x = 0, y = - 5, z = - 3$ .
1½ marks

Answer: $A^{- 1} = - 3 - 2 - 4 212213$ ; x = 0, y = −5, z = −3

Where the marks are

|A| = 1 ≠ 0 (1 mark)
Correct adj A (2 marks)
Writes A⁻¹ (0.5 mark)
Writes AX = B and solves X = A⁻¹B to get x = 0, y = −5, z = −3 (1.5 marks)

Revise the chapter

Chapter: Matrices (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Matrices, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 32(a)

Hints

Step-by-step solution

Revise the chapter

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