Find the product of the matrices bmatrix1&2&-3\\2&3&2\\3&-3&-4bmatrixbmatrix-6&17&13\\14&5&-8\\-15&9&-1bmatrix and hence solve the system of linear equations : x+2y-3z=-4 2x+3y+2z=2 3x-3y-4z=11

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

CBSE Class 12 Mathematics 2024: question 32(b)

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

Q32(b) · 5 marks · Long answerMatrices

Find the product of the matrices $123 23 - 3 - 3 2 - 4 - 6 14 - 15 1759 13 - 8 - 1$ and hence solve the system of linear equations :
$x + 2 y - 3 z = - 4$
$2 x + 3 y + 2 z = 2$
$3 x - 3 y - 4 z = 11$

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/5/1.

Hints

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

Hint 1

Multiply carefully row by column; the answer is a scalar multiple of I.

Hint 2

If $A B = k I$ , then $A^{- 1} = \frac{1}{k} B$ .

Hint 3

Write the system as $A X = C$ and use $X = A^{- 1} C$ .

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.

Let $A = 123 23 - 3 - 3 2 - 4$ , $B = - 6 14 - 15 1759 13 - 8 - 1$ . Row 1: $- 6 + 28 + 45 = 67$ , $17 + 10 - 27 = 0$ , $13 - 16 + 3 = 0$ ; Row 2: $- 12 + 42 - 30 = 0$ , $34 + 15 + 18 = 67$ , $26 - 24 - 2 = 0$ ; Row 3: $- 18 - 42 + 60 = 0$ , $51 - 15 - 36 = 0$ , $39 + 24 + 4 = 67$ . So $A B = 670006700067 = 67 I$ .
2½ marks
$A (\frac{1}{67} B) = I \Rightarrow A^{- 1} = \frac{1}{67} - 6 14 - 15 1759 13 - 8 - 1$ .
1 mark
The system is $A X = C$ , $C = - 4 211$ . $X = A^{- 1} C = \frac{1}{67} 24 + 34 + 143 - 56 + 10 - 88 60 + 18 - 11 = \frac{1}{67} 201 - 134 67 = 3 - 2 1$ . Hence $x = 3$ , $y = - 2$ , $z = 1$ .
1½ marks

Answer: Product $= 67 I$ ; $x = 3$ , $y = - 2$ , $z = 1$

Where the marks are

Correct product $67 I$ (2½ marks)
Deduces $A^{- 1} = \frac{1}{67} B$ (1 mark)
Solves $X = A^{- 1} C$ to get $x = 3, y = - 2, z = 1$ (1½ 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 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 32(b)

Hints

Step-by-step solution

Revise the chapter

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