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

The Correct Option is A

Given Matrix A: 

$A = \begin{bmatrix} 2 & 4 & 5 \\ 0 & 7 & 4 \\ 3 & 11 & 8 \end{bmatrix}$

To find the determinant of a 3x3 matrix, we use:

$\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$

Where the matrix is structured as:

$\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$

From matrix A, we identify:

• $a = 2$, $b = 4$, $c = 5$
• $d = 0$, $e = 7$, $f = 4$
• $g = 3$, $h = 11$, $i = 8$

Now apply the formula:

$\text{det}(A) = 2(7 \cdot 8 - 4 \cdot 11) - 4(0 \cdot 8 - 4 \cdot 3) + 5(0 \cdot 11 - 7 \cdot 3)$

Step-by-step:

• $7 \cdot 8 = 56$
• $4 \cdot 11 = 44$
• $0 \cdot 8 = 0$
• $4 \cdot 3 = 12$
• $0 \cdot 11 = 0$
• $7 \cdot 3 = 21$

So we get:

$\text{det}(A) = 2(56 - 44) - 4(0 - 12) + 5(0 - 21)$

$= 2(12) - 4(-12) + 5(-21)$

$= 24 + 48 - 105 = -33$

Check divisibility:

$-33$ is divisible by $11$, since $-33 = 11 \cdot (-3)$

Final Answer: Option (A): det A is divisible by 11

Answer 2

Step 1: Calculate the Determinant of A

The determinant of a 3x3 matrix is calculated as:

\(|A| = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\)

Plugging in the values from matrix A:

\(|A| = 2(7 \cdot 8 - 11 \cdot 4) - 0(4 \cdot 8 - 11 \cdot 5) + 3(4 \cdot 4 - 7 \cdot 5)\)

\(|A| = 2(56 - 44) - 0 + 3(16 - 35)\)

\(|A| = 2(12) + 3(-19) = 24 - 57 = -33\)

Step 2: Check Divisibility by 11

Since \(|A| = -33\), it is divisible by 11 because \(-33 = -3 \cdot 11\).

Step 3: Check if det A = 0

\(|A| = -33 \neq 0\), so det A is not 0.

Step 4: Check if A is an Orthogonal Matrix

For a matrix to be orthogonal, its transpose multiplied by itself must equal the identity matrix, i.e., \(AA^T = I\). In the above case this doesn't hold therefore it isn't orthogonal

\(A^T = \begin{pmatrix} 2 & 4 & 5 \\ 0 & 7 & 4 \\ 3 & 11 & 8 \end{pmatrix}\)

\({A^T} \) This option implies it.

Conclusion:

Since the determinant of A is -33 and is divisible by 11, the correct statement is:

det A is divisible by 11.