More practice questions

5. Q5. Simplify cos θ times the matrix with rows cos θ, sin θ and −sin θ, cos θ, plus sin θ times the matrix with rows sin θ, −cos θ and cos θ, sin θ (NCERT Exercise 3.2, Q6).

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   Position by position: cos² θ + sin² θ = 1, cos θ sin θ − sin θ cos θ = 0, −sin θ cos θ + sin θ cos θ = 0 and cos² θ + sin² θ = 1. The sum is I.

6. Q6. Find X if Y has rows 3, 2 and 1, 4, and 2X + Y has rows 1, 0 and −3, 2 (NCERT Exercise 3.2, Q8).

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   2X = (2X + Y) − Y has rows −2, −2 and −4, −2, so X has rows −1, −1 and −2, −1.

7. Q7. Solve for x, y, z and t (NCERT Exercise 3.2, Q10).

   $$\begin{aligned} &2\begin{bmatrix} x & z \\ y & t \end{bmatrix} + 3\begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix} \\ &= 3\begin{bmatrix} 3 & 5 \\ 4 & 6 \end{bmatrix} \end{aligned}$$
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   2x + 3 = 9, 2z − 3 = 15, 2y = 12 and 2t + 6 = 18, so x = 3, z = 9, y = 6 and t = 6.

8. Q8. If x times the column (2, 3) plus y times the column (−1, 1) equals the column (10, 5), find x and y (NCERT Exercise 3.2, Q11).

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   2x − y = 10 and 3x + y = 5. Adding, 5x = 15, so x = 3 and y = −4.

9. Q9. Show that the two products below differ (NCERT Exercise 3.2, Q14 (i)).

   $$\begin{bmatrix} 5 & -1 \\ 6 & 7 \end{bmatrix}\begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$$ $$\begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}\begin{bmatrix} 5 & -1 \\ 6 & 7 \end{bmatrix}$$
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   The first has rows 10 − 3, 5 − 4 and 12 + 21, 6 + 28, that is 7, 1 and 33, 34. The second has rows 10 + 6, −2 + 7 and 15 + 24, −3 + 28, that is 16, 5 and 39, 25. They differ.

10. Q10. If A has rows 1, 0, 2; 0, 2, 1; 2, 0, 3, prove that A³ − 6A² + 7A + 2I = O (NCERT Exercise 3.2, Q16).

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    A² has rows 5, 0, 8; 2, 4, 5; 8, 0, 13, and A³ = A A² has rows 21, 0, 34; 12, 8, 23; 34, 0, 55. Then A³ − 6A² + 7A + 2I has (1, 1) element 21 − 30 + 7 + 2 = 0, and every other element works out to 0 in the same way, so the sum is O. NCERT slip (page 60): the question prints the right side as 0; it means the zero matrix O. In an exam, write O.

11. Q11. For F(x) with rows cos x, −sin x, 0; sin x, cos x, 0; 0, 0, 1, show that F(x) F(y) = F(x + y) (NCERT Exercise 3.2, Q13).

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    The (1, 1) element of the product is cos x cos y − sin x sin y = cos(x + y), the (1, 2) element is −cos x sin y − sin x cos y = −sin(x + y), the (2, 1) element is sin(x + y) and the (2, 2) element is cos(x + y), by the sum formulas; the last row and column stay 0, 0, 1. That is F(x + y).

12. Q12. A political group spends 40 paise per telephone call, 100 paise per house call and 50 paise per letter. In city X it makes 1000 calls, 500 house calls and 5000 letters; in city Y, 3000, 1000 and 10 000. Find the amount spent in each city (NCERT Example 19, pages 57–58).

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    X: 1000 × 40 + 500 × 100 + 5000 × 50 = 340 000 paise = ₹3400. Y: 3000 × 40 + 1000 × 100 + 10 000 × 50 = 720 000 paise = ₹7200.

13. Q13. A is a square matrix with A² = A. Then (I + A)³ − 7A equals (A) A; (B) I − A; (C) I; (D) 3A (NCERT Miscellaneous Exercise, Q11, page 73).

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    I and A commute, so (I + A)³ = I + 3A + 3A² + A³. With A² = A, also A³ = A, so (I + A)³ = I + 7A, and (I + A)³ − 7A = I: (C).

14. Q14. Find a matrix D with CD − AB = O, where C has rows 2, 5 and 3, 8; A has rows 2, −1 and 3, 4; and B has rows 5, 2 and 7, 4 (NCERT Miscellaneous Example 25, pages 71–72).

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    AB has rows 3, 0 and 43, 22. With D having rows a, b and c, d: 2a + 5c = 3 and 3a + 8c = 43 give a = −191, c = 77; 2b + 5d = 0 and 3b + 8d = 22 give b = −110, d = 44.

15. Q15. For what value of x is the row [1, 2, 1] times the matrix with rows 1, 2, 0; 2, 0, 1; 1, 0, 2 times the column with entries 0, 2, x equal to O? (NCERT Miscellaneous Exercise, Q4, page 72)

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    The row times the matrix gives [1 + 4 + 1, 2 + 0 + 0, 0 + 2 + 2] = [6, 2, 4]. Times the column: 0 + 4 + 4x = 0, so x = −1.

16. Q16. Find x if the row [x, −5, −1] times the matrix with rows 1, 0, 2; 0, 2, 1; 2, 0, 3 times the column with entries x, 4, 1 is O (NCERT Miscellaneous Exercise, Q6, page 72).

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    The row times the matrix gives [x − 2, −10, 2x − 8]. Times the column: x² − 2x − 40 + 2x − 8 = x² − 48 = 0, so x = ±4√3.

17. Q17. A manufacturer sells products x, y, z in two markets: market I takes 10 000, 2000 and 18 000 units, market II 6000, 20 000 and 8000. Unit prices are ₹2.50, ₹1.50 and ₹1.00, and unit costs ₹2.00, ₹1.00 and 50 paise. Find the revenue and the gross profit in each market (NCERT Miscellaneous Exercise, Q7, page 72).

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    Revenue: market I, 10 000 × 2.50 + 2000 × 1.50 + 18 000 × 1.00 = ₹46 000; market II, 6000 × 2.50 + 20 000 × 1.50 + 8000 × 1.00 = ₹53 000. Cost: I, 20 000 + 2000 + 9000 = ₹31 000; II, 12 000 + 20 000 + 4000 = ₹36 000. Gross profit: ₹15 000 and ₹17 000.