Question

Let

\[ A = \begin{bmatrix} a & u_1 & u_2 & u_3 \end{bmatrix}, \quad B = \begin{bmatrix} b & u_1 & u_2 & u_3 \end{bmatrix}, \quad C = \begin{bmatrix} u_2 & u_3 & u_1 & a + b \end{bmatrix}. \]

Let \( \det(A), \det(B), \det(C) \) denote the determinants of the matrices \( A \), \( B \), and \( C \), respectively. If

\[ \det(A) = 6, \quad \det(B) = 2, \quad \text{then } \det(A + B) - \det(C) = \underline{\hspace{2cm}}. \]

Hint:

When working with determinants of sums, use properties of determinants and matrix operations to simplify the calculations.

Answer 1

Correct Answer: 72

Given the determinants of \( A \) and \( B \), we can use properties of determinants to simplify the expression \( \det(A + B) - \det(C) \). Calculating gives: \[ \det(A + B) - \det(C) = 7. \] Thus, the value is \( 7 \).