Questions: Consider the following matrix [ 3 4 0 3 0 3 0 1 -2 ] (a) Find its determinant. (b) Does the matrix have an inverse?

Transcript text: Consider the following matrix \[ \left[\begin{array}{ccc} 3 & 4 & 0 \\ 3 & 0 & 3 \\ 0 & 1 & -2 \end{array}\right] \] (a) Find its determinant. (b) Does the matrix have an inverse? Choose Submit answer Next item Answers

Solution

Solution Steps

To solve the given problems, we will first calculate the determinant of the matrix. If the determinant is non-zero, the matrix has an inverse; otherwise, it does not.

Step 1: Calculate the Determinant

To determine if the matrix has an inverse, we first calculate its determinant. The given matrix is:

\[ \begin{bmatrix} 3 & 4 & 0 \\ 3 & 0 & 3 \\ 0 & 1 & -2 \end{bmatrix} \]

The determinant of a \(3 \times 3\) matrix \(\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\) is calculated as:

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

Applying this formula to our matrix:

\[ \text{det}(A) = 3(0 \cdot (-2) - 3 \cdot 1) - 4(3 \cdot (-2) - 3 \cdot 0) + 0(3 \cdot 1 - 0 \cdot 0) \]

\[ = 3(0 - 3) - 4(-6) + 0 \]

\[ = 3(-3) + 24 \]

\[ = -9 + 24 \]

\[ = 15 \]

Step 2: Determine if the Matrix has an Inverse

A matrix has an inverse if and only if its determinant is non-zero. Since the determinant of the matrix is \(15\), which is non-zero, the matrix does have an inverse.