More practice questions

5. Q5. A has rows α, β and γ, −α, and A² = I. Then (A) 1 + α² + βγ = 0; (B) 1 − α² + βγ = 0; (C) 1 − α² − βγ = 0; (D) 1 + α² − βγ = 0 (NCERT Miscellaneous Exercise, Q9, page 73).

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   A² has rows α² + βγ, αβ − βα and γα − αγ, γβ + α², that is (α² + βγ)I. So A² = I exactly when α² + βγ = 1: (C).

6. Q6. Let A have rows cos θ, sin θ and −sin θ, cos θ. NCERT Miscellaneous Example 23 (page 70) proves by induction that, for every natural number n, A^ⁿ has rows cos nθ, sin nθ and −sin nθ, cos nθ. Find A^⁻¹ and check that it fits the same pattern with n = −1.

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   Try the pattern with n = −1: rows cos θ, −sin θ and sin θ, cos θ. Its product with A, in either order, has rows cos² θ + sin² θ, 0 and 0, cos² θ + sin² θ, which is I. So A^⁻¹ is this matrix, which is also A′.

7. Q7. Use the 2 × 2 shortcut to find the inverse of the matrix with rows 4, −3 and 2, −1.

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   ad − bc = −4 + 6 = 2, so the inverse is ½ times the matrix with rows −1, 3 and −2, 4, that is rows −1/2, 3/2 and −1, 2. Check: the first row of A times the columns gives −2 + 3 = 1 and 6 − 6 = 0.

8. Q8. Find the inverse of the diagonal matrix with diagonal 2, −5, 1/3. When does a diagonal matrix have an inverse?

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   The inverse is diagonal with the reciprocals 1/2, −1/5 and 3, since diagonal matrices multiply element by element on the diagonal. A diagonal matrix is invertible exactly when none of its diagonal elements is 0.

9. Q9. If A, B and C are invertible matrices of the same order, show that (ABC)^⁻¹ = C^⁻¹B^⁻¹A^⁻¹.

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   (ABC)(C^⁻¹B^⁻¹A^⁻¹) = AB(CC^⁻¹)B^⁻¹A^⁻¹ = A(BB^⁻¹)A^⁻¹ = AA^⁻¹ = I, and in the same way (C^⁻¹B^⁻¹A^⁻¹)(ABC) = I.