Question

If $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 1 \\ 1 & 3 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 3 & 4 \\ 3 & 2 & 2 \\ 2 & 4 & 2 \end{pmatrix}$, then $\sqrt{|\text{Adj}(AB)|} =$

• A. 176
• B. 208
• C. 198
• D. 234

Hint:

Remember the key properties of determinants and adjoints: $|\text{Adj}(M)| = |M|^{n-1}$ and $|AB| = |A||B|$. These properties save a significant amount of time compared to calculating the matrix product AB and then its adjoint.

Answer 1

The Correct Option is C

To find the value of \(\sqrt{|\text{Adj}(AB)|}\), where \(A\) and \(B\) are given matrices, we can follow these steps:

Step 1: Determinant of AB

The determinant of the product of two matrices is the product of the determinants of the individual matrices.

• \(|AB| = |A| \times |B|\)

Step 2: Determinant of A

Calculate the determinant of matrix \(A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 1 \\ 1 & 3 & 1 \end{pmatrix}\).

• Using co-factor expansion along the first row:

\(|A| = 1 \cdot \begin{vmatrix} 1 & 1 \\ 3 & 1 \end{vmatrix} - 2 \cdot \begin{vmatrix} 2 & 1 \\ 1 & 1 \end{vmatrix} + 3 \cdot \begin{vmatrix} 2 & 1 \\ 1 & 3 \end{vmatrix}\)

• \(|A| = 1 \cdot (1 \cdot 1 - 1 \cdot 3) - 2 \cdot (2 \cdot 1 - 1 \cdot 1) + 3 \cdot (2 \cdot 3 - 1 \cdot 1)\)
• \(= 1(-2) - 2(1) + 3(5) = -2 - 2 + 15 = 11\)

Step 3: Determinant of B

Calculate the determinant of matrix \(B = \begin{pmatrix} 2 & 3 & 4 \\ 3 & 2 & 2 \\ 2 & 4 & 2 \end{pmatrix}\).

• Using co-factor expansion along the first row:

\(|B| = 2 \cdot \begin{vmatrix} 2 & 2 \\ 4 & 2 \end{vmatrix} - 3 \cdot \begin{vmatrix} 3 & 2 \\ 2 & 2 \end{vmatrix} + 4 \cdot \begin{vmatrix} 3 & 2 \\ 2 & 4 \end{vmatrix}\)

• \(|B| = 2(2 \cdot 2 - 4 \cdot 2) - 3(3 \cdot 2 - 2 \cdot 2) + 4(3 \cdot 4 - 2 \cdot 2)\)
• \(= 2(4 - 8) - 3(6 - 4) + 4(12 - 4)\)
• \(= 2(-4) - 3(2) + 4(8) = -8 - 6 + 32 = 18\)

Step 4: Determinant of Product AB

Now calculate the determinant of \(AB\):

• \(|AB| = |A| \times |B| = 11 \times 18 = 198\)

Step 5: Determinant of Adj(AB)

The determinant of the adjugate of a matrix is given by:

• \(|\text{Adj}(AB)| = |AB|^{n-1}\)

Since \(AB\) is a 3x3 matrix, \(n = 3\), thus:

• \(|\text{Adj}(AB)| = |AB|^2 = 198^2\)

Step 6: Calculating Square Root

We need to find \(\sqrt{|\text{Adj}(AB)|}\):

• \(\sqrt{|AB|^2} = |AB| = 198\)

Thus, the correct answer is 198.