$$ A=\left[\begin{array}{ccc}-8 & -8 & 3 \ 8 & 5 & 5 \ -7 & -2 & -2\end{array}\right] $$ using the method of Minor's by expanding along the first row. The formula for det $$ A $$ using the method of Minor's by expanding along the first row for the matrix $$ A $$ above is, $$ \operatorname{det} A=\left(-8 \times \operatorname{det} A_{11}\right)-\left(-8 \times \operatorname{det} A_{12}\right)+\left(3 \times \operatorname{det} A_{13}\right) $$ for 3 appropriate submatrices $$ A_{11}, A_{12} $$, and $$ A_{13} $$ of the matrix $$ A $$.

Question

UpStudy AI · Accepted Answer

We are given the matrix

$$
A=\begin{bmatrix} -8 & -8 & 3 \ 8 & 5 & 5 \ -7 & -2 & -2 \end{bmatrix}
$$

and we want to compute its determinant by expanding along the first row using minors.

**Step 1. Write the expansion formula:**

$$
\det A = a_{11}\,\det A_{11} - a_{12}\,\det A_{12} + a_{13}\,\det A_{13}
$$

where $$a_{11}=-8$$, $$a_{12}=-8$$, and $$a_{13}=3$$.

**Step 2. Determine the 2x2 submatrices:**

- To form $$A_{11}$$, remove the first row and the first column:

$$
A_{11}=\begin{bmatrix} 5 & 5 \ -2 & -2 \end{bmatrix}
$$

- To form $$A_{12}$$, remove the first row and the second column:

$$
A_{12}=\begin{bmatrix} 8 & 5 \ -7 & -2 \end{bmatrix}
$$

- To form $$A_{13}$$, remove the first row and the third column:

$$
A_{13}=\begin{bmatrix} 8 & 5 \ -7 & -2 \end{bmatrix}
$$

**Step 3. Compute the determinants of the submatrices:**

For $$A_{11}$$:

$$
\det A_{11} = (5)(-2) - (5)(-2) = -10 + 10 = 0
$$

For $$A_{12}$$:

$$
\det A_{12} = (8)(-2) - (5)(-7) = -16 + 35 = 19
$$

For $$A_{13}$$:

$$
\det A_{13} = (8)(-2) - (5)(-7) = -16 + 35 = 19
$$

**Step 4. Substitute back into the expansion:**

$$
\det A = (-8)(0) - (-8)(19) + (3)(19)
$$

Simplify each term:

- First term: $$-8 \times 0 = 0$$
- Second term: $$-(-8 \times 19) = +8 \times 19 = 152$$
- Third term: $$3 \times 19 = 57$$

**Step 5. Sum the terms:**

$$
\det A = 0 + 152 + 57 = 209
$$

Thus, the determinant of $$ A $$ is

$$
\boxed{209}
$$
The determinant of matrix $$ A $$ is 209.

using the method of Minor’s by expanding along the first row.
The formula for det using the method of Minor’s by expanding along the first row for the matrix above
is,

for 3 appropriate submatrices , and of the matrix .

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Calculus Questions

using the method of Minor’s by expanding along the first row. The formula for det using the method of Minor’s by expanding along the first row for the matrix above is, for 3 appropriate submatrices , and of the matrix .