Question

For a given square matrix P, of order q, if $|adj(P)|=|P|^k$, what is the value of k?

• A. k = 3q
• B. k = q
• C. k = q - 1
• D. k = q + 1

Answer 1

Answer: (C)

k = q - 1

Answer 2

To find the value of k for a given square matrix P of order q, such that $|adj(P)|=|P|^k$, we use the fundamental property relating a matrix, its adjoint, and its determinant.

1. Fundamental Property of Adjoint Matrix:

For any square matrix P of order q, the product of the matrix and its adjoint is equal to the determinant of the matrix times the identity matrix of the same order:

$P \cdot adj(P) = |P| I_q$

where $I_q$ is the identity matrix of order q.

2. Taking Determinant on Both Sides:

Now, we take the determinant of both sides of the equation:

$|P \cdot adj(P)| = ||P| I_q|$

3. Applying Determinant Properties:

We use two properties of determinants:

• $|AB| = |A||B|$ for square matrices A and B.
• $|cA| = c^n |A|$ for a scalar c and a square matrix A of order n. In our case, $A = I_q$ and $n=q$. Also, $|I_q|=1$. So, $||P| I_q| = (|P|)^q |I_q| = (|P|)^q \cdot 1 = (|P|)^q$.

Applying these properties to our equation:

$|P| \cdot |adj(P)| = (|P|)^q$

4. Substituting the Given Relation:

We are given that $|adj(P)| = |P|^k$. Substitute this into the equation:

$|P| \cdot |P|^k = (|P|)^q$

5. Simplifying the Equation:

Using the rule of exponents $a^m \cdot a^n = a^{m+n}$:

$|P|^{1+k} = |P|^q$

6. Solving for k:

This equation holds true for all square matrices.

• Case 1: If $|P| \neq 0$ (P is a non-singular matrix):

If the base $|P|$ is non-zero, we can equate the exponents:

$1+k = q$

$k = q-1$

• Case 2: If $|P| = 0$ (P is a singular matrix):

The equation becomes $0^{1+k} = 0^q$.

If $q > 1$, then $q-1 > 0$. In this case, $0^q = 0$ and $0^{1+k} = 0$ (since $1+k = q > 0$). So, $0=0$, which is consistent with $k=q-1$.

The only exception is for $q=1$ and $|P|=0$. For a $1 \times 1$ matrix $P=[0]$, $adj(P)=[1]$. So $|adj(P)|=1$. The given relation becomes $1 = 0^k$, which is impossible for any $k$. However, such questions typically refer to the general property derived from the non-singular case, which is universally accepted in matrix theory.

Therefore, the value of k that satisfies the relation for a general square matrix is $k=q-1$.