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

Concept:
A square matrix is said to be singular if its determinant is zero. A singular matrix does not have an inverse.

Step 1: Write the given matrix: \[ A=\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \]

Step 2: Observe the rows of the matrix. The rows are not independent. We can see that: \[ R_2-R_1 = [4-1,\;5-2,\;6-3]=[3,3,3] \] and \[ R_3-R_2 = [7-4,\;8-5,\;9-6]=[3,3,3] \]

Step 3: Since the row differences are the same, the rows are linearly dependent.

Step 4: If rows of a square matrix are linearly dependent, then determinant of the matrix is zero. \[ |A|=0 \]

Step 5: A matrix whose determinant is zero is called a singular matrix. \[ \boxed{\text{The matrix is singular}} \]