Question

Let $A=\begin{pmatrix}2&0&3\\\4&7&11\\\5&4&8\end{pmatrix}$. Then

• A. det A is divisible by 11
• B. det A is not divisible by 11
• C. det A=0
• D. A is orthogonal matrix

Answer 1

Answer: (A)

det A is divisible by 11

Answer 2

We're given matrix A, which is a 3x3 matrix.

The determinant of a 3x3 matrix can be calculated using the following formula:

det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

Here, a, b, c, d, e, f, g, h, i are the elements of the 3x3 matrix A in the following arrangement:

[ a b c ] [d e f] [g h i ]

We're given the values of matrix A:

A = [2 4 5 0 7 4 3 11 8 ]

Plugging these values into the determinant formula:

det(A) = 2(7 * 8 - 4 * 11) - 4(0 * 8 - 4 * 3) + 5(0 * 11 - 7 * 3)

Calculating the determinant:

det(A) = 2(56 - 44) - 4(0 - 12) + 5(0 - 21) = 2(12) - 4(-12) + 5(-21) = 24 + 48 - 105 = -33

We can see that the determinant of matrix A is -33, which is divisible by 11 (since -33 = 11 * (-3)).

Therefore, the answer is correct, and the determinant of matrix A is indeed divisible by 11.

The correct answer is option (A): det A is divisible by 11