Questions: The value of k which makes the matrix [9 -2 -3; -8 1 -8; 7 k 4] singular is k =

The value of k which makes the matrix [9 -2 -3; -8 1 -8; 7 k 4] singular is k =
Transcript text: 4. Submit answer Get help Practice similar Attempt 6: 5 attempts remaining. The value of $k$ which makes the matrix $\left[\begin{array}{ccc}9 & -2 & -3 \\ -8 & 1 & -8 \\ 7 & k & 4\end{array}\right]$ singular is $k=$ $\square$ Submit answer Next item Answers Attempt 6 of 6
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Solution

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Solution Steps

To determine the value of \( k \) that makes the matrix singular, we need to find the value of \( k \) for which the determinant of the matrix is zero. A matrix is singular if and only if its determinant is zero.

Step 1: Define the Matrix and Calculate the Determinant

We start with the matrix: \[ \begin{bmatrix} 9 & -2 & -3 \\ -8 & 1 & -8 \\ 7 & k & 4 \end{bmatrix} \] To find the value of \( k \) that makes the matrix singular, we need to calculate its determinant and set it to zero.

Step 2: Set Up the Determinant Equation

The determinant of the matrix is given by: \[ \text{det} = 9 \left(1 \cdot 4 - (-8) \cdot k\right) - (-2) \left(-8 \cdot 4 - (-8) \cdot 7\right) + (-3) \left(-8 \cdot k - 1 \cdot 7\right) \] Simplifying this, we get: \[ \text{det} = 9 (4 + 8k) - (-2) (32 + 56) + (-3) (-8k - 7) \] \[ \text{det} = 9 (4 + 8k) + 2 (88) + 3 (8k + 7) \] \[ \text{det} = 36 + 72k + 176 + 24k + 21 \] \[ \text{det} = 96k + 233 \]

Step 3: Solve for \( k \)

To make the matrix singular, we set the determinant to zero: \[ 96k + 233 = 0 \] Solving for \( k \): \[ k = -\frac{233}{96} \] \[ k = -\frac{35}{32} \]

Final Answer

The value of \( k \) that makes the matrix singular is: \[ \boxed{k = -\frac{35}{32}} \]

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