Question

If the matrix \[ A=\begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}, \] then which of the following is true?

• A. The matrix is invertible
• B. The matrix is singular
• C. The matrix is diagonalizable
• D. The matrix is symmetric

Hint:

If the determinant of a square matrix is zero, then the matrix is singular and has no inverse.

Answer 1

The Correct Option is B

To determine whether the given matrix \(A=\begin{bmatrix}1 & 2 & 3\\4 & 5 & 6\\7 & 8 & 9\end{bmatrix}\) is singular, we need to calculate its determinant. A matrix is singular if its determinant is zero. Let's calculate the determinant:

1. Write down the formula for the determinant of a 3x3 matrix \(\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}:\) \(\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\).
2. Substitute the elements of matrix \(A\) into the formula: \(\det(A) = 1(5 \cdot 9 - 6 \cdot 8) - 2(4 \cdot 9 - 6 \cdot 7) + 3(4 \cdot 8 - 5 \cdot 7)\).
3. Calculate each term:
   • \(5 \cdot 9 - 6 \cdot 8 = 45 - 48 = -3\)
   • \(4 \cdot 9 - 6 \cdot 7 = 36 - 42 = -6\)
   • \(4 \cdot 8 - 5 \cdot 7 = 32 - 35 = -3\)
4. Plug these values back into the determinant formula: \(\det(A) = 1(-3) - 2(-6) + 3(-3)\).
5. Simplify to find the determinant:
   • \(1(-3) = -3\)
   • \(- 2(-6) = 12\)
   • \(+ 3(-3) = -9\)
6. Calculate the final result: \(-3 + 12 - 9 = 0\).

Since the determinant is 0, the matrix \(A\) is singular.

Option analysis:

• The matrix is invertible: False, as the determinant is 0, indicating it is not invertible.
• The matrix is singular: True, the correct answer since the determinant is 0.
• The matrix is diagonalizable: Not necessarily true for singular matrices.
• The matrix is symmetric: False, as the matrix is not equal to its transpose.

The correct answer is: The matrix is singular.