Question

If P and Q are two invertible matrices of the same order, then adj (PQ) is equal to

• A. adj (Q) adj (P)
• B. |Q||P|Q$^{-1}$P$^{-1}$
• C. |Q||P|P$^{-1}$Q$^{-1}$
• D. |P||Q|(PQ)$^{-1}$

Answer 1

Answer: (A)

adj (Q) adj (P)

Answer 2

To find adj(PQ) for invertible matrices P and Q of the same order, we use the properties of determinants, inverses, and adjoints.

1. Definition of Adjoint: For an invertible matrix A, its adjoint is given by adj(A) = |A| A⁻¹. Applying this definition to PQ: adj(PQ) = |PQ| (PQ)⁻¹

2. Determinant of a Product: The determinant of a product of matrices is the product of their determinants: |PQ| = |P| |Q|

3. Inverse of a Product: The inverse of a product of matrices is the product of their inverses in reverse order: (PQ)⁻¹ = Q⁻¹ P⁻¹

4. Substitute the properties: Substitute the expressions from steps 2 and 3 into the equation from step 1: adj(PQ) = (|P| |Q|) (Q⁻¹ P⁻¹) adj(PQ) = |P| |Q| Q⁻¹ P⁻¹

The adjoint of a product of two invertible matrices, P and Q, is given by the product of their adjoints in reverse order. This is a standard property: adj(PQ) = adj(Q) adj(P). Alternatively, using the definition adj(A) = |A| A⁻¹ and properties |AB| = |A||B| and (AB)⁻¹ = B⁻¹A⁻¹: adj(PQ) = |PQ|(PQ)⁻¹ = |P||Q|Q⁻¹P⁻¹. Since adj(Q)adj(P) = (|Q|Q⁻¹)(|P|P⁻¹) = |Q||P|Q⁻¹P⁻¹, both expressions are equivalent.