More practice questions

5. Q5. For A with rows 1, 5 and 6, 7, verify that A + A′ is symmetric and A − A′ is skew symmetric (NCERT Exercise 3.3, Q8).

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   A′ has rows 1, 6 and 5, 7. A + A′ has rows 2, 11 and 11, 14, equal to its transpose. A − A′ has rows 0, −1 and 1, 0, and its transpose has rows 0, 1 and −1, 0, which is −(A − A′).

6. Q6. Express each matrix as the sum of a symmetric and a skew symmetric matrix: (i) rows 6, −2, 2; −2, 3, −1; 2, −1, 3; (ii) rows 1, 5 and −1, 2 (NCERT Exercise 3.3, Q10 (ii) and (iv)).

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   (i) The matrix is already symmetric (A′ = A), so it is A + O, with the zero matrix as the skew symmetric part. (ii) A′ has rows 1, −1 and 5, 2. ½(A + A′) has rows 1, 2 and 2, 2; ½(A − A′) has rows 0, 3 and −3, 0. Their sum is A.

7. Q7. A has rows cos α, −sin α and sin α, cos α, and A + A′ = I. Then α is (A) π/6; (B) π/3; (C) π; (D) 3π/2 (NCERT Exercise 3.3, Q12).

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   A + A′ has rows 2 cos α, 0 and 0, 2 cos α. Setting this equal to I gives cos α = 1/2, and of the options only π/3 works: (B).

8. Q8. If the matrix A is both symmetric and skew symmetric, then (A) A is a diagonal matrix; (B) A is a zero matrix; (C) A is a square matrix; (D) none of these (NCERT Miscellaneous Exercise, Q10, page 73).

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   a_ij = a_ji and a_ji = −a_ij give a_ij = 0 for every i and j: (B). (A) and (C) are also true of the zero matrix, but (B) is the best answer, as in NCERT’s key.

9. Q9. Find x, y and z if the matrix below is skew symmetric.

   $$\begin{bmatrix} 0 & x+2 & 4 \\ -5 & 0 & y \\ z & 3 & 0 \end{bmatrix}$$
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   Skew symmetry needs a₂₁ = −a₁₂: −5 = −(x + 2), so x = 3. Also a₃₁ = −a₁₃ gives z = −4, and a₃₂ = −a₂₃ gives 3 = −y, so y = −3.

10. Q10. If A and B are symmetric matrices of the same order, show that AB + BA is symmetric.

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    (AB + BA)′ = (AB)′ + (BA)′ = B′A′ + A′B′ = BA + AB. So AB + BA equals its transpose.