Use a determinant to find the value of $$ a $$ that make the matrix $$ A=\left[\begin{array}{ccc}a & 2 & 2 \ -1 & 2 & 3 \ 3 & 1 & 2\end{array} ight] $$ singular. Q Question Help: $$ \square $$ Video Submit Question

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Use a determinant to find the value of $$ a $$ that make the matrix $$ A=\left[\begin{array}{ccc}a & 2 & 2 \ -1 & 2 & 3 \ 3 & 1 & 2\end{array}ight] $$ singular. Q Question Help: $$ \square $$ Video Submit Question

UpStudy AI · Accepted Answer

1. We are given the matrix

$$
   A = \begin{pmatrix}
   a & 2 & 2 \
   -1 & 2 & 3 \
   3 & 1 & 2
   \end{pmatrix}.
   $$

The matrix is singular when its determinant is zero.

2. Compute the determinant using cofactor expansion along the first row:

$$
   \det(A) = a \cdot \det\begin{pmatrix} 2 & 3 \ 1 & 2 \end{pmatrix} - 2 \cdot \det\begin{pmatrix} -1 & 3 \ 3 & 2 \end{pmatrix} + 2 \cdot \det\begin{pmatrix} -1 & 2 \ 3 & 1 \end{pmatrix}.
   $$

3. Calculate each $$2 \times 2$$ determinant:

- For the first minor:

$$
     \det\begin{pmatrix} 2 & 3 \ 1 & 2 \end{pmatrix} = 2 \cdot 2 - 3 \cdot 1 = 4 - 3 = 1.
     $$

- For the second minor:

$$
     \det\begin{pmatrix} -1 & 3 \ 3 & 2 \end{pmatrix} = (-1) \cdot 2 - 3 \cdot 3 = -2 - 9 = -11.
     $$

- For the third minor:

$$
     \det\begin{pmatrix} -1 & 2 \ 3 & 1 \end{pmatrix} = (-1) \cdot 1 - 2 \cdot 3 = -1 - 6 = -7.
     $$

4. Substitute these results into the determinant formula:

$$
   \det(A) = a(1) - 2(-11) + 2(-7) = a + 22 - 14 = a + 8.
   $$

5. Set the determinant equal to zero and solve for $$ a $$:

$$
   a + 8 = 0 \quad \Longrightarrow \quad a = -8.
   $$

Thus, the value of $$ a $$ that makes the matrix singular is $$ a = -8 $$.
The matrix $$ A $$ is singular when $$ a = -8 $$.

Use a determinant to find the value of that make the matrix singular.
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