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Problem 139

# Evaluate det $$\mathrm{A}$$ where: $A=\left|\begin{array}{rrr}0 & 1 & 5 \\ 3 & -6 & 9 \\ 2 & 6 & 1\end{array}\right|$

Expert verified
The determinant of matrix $$A = \begin{pmatrix} 0 & 1 & 5 \\ 3 & -6 & 9 \\ 2 & 6 & 1 \end{pmatrix}$$ can be found using the expansion by minors. First, expand along the first row: $$\det(A) = 0 \cdot \det(M_{11}) - 1 \cdot \det(M_{12}) + 5 \cdot \det(M_{13})$$, where the minor matrices are $$M_{11} = \begin{pmatrix}-6 & 9 \\ 6 & 1 \end{pmatrix}$$, $$M_{12} = \begin{pmatrix} 3 & 9 \\ 2 & 1 \end{pmatrix}$$, and $$M_{13} = \begin{pmatrix} 3 & -6 \\ 2 & 6 \end{pmatrix}$$. Then, find the determinant of each minor matrix: $$\det(M_{11}) = -60$$, $$\det(M_{12}) = -15$$, and $$\det(M_{13}) = 24$$. Substituting these values back into the determinant formula, we get $$\det(A) = 0 + 15 + 120 = 135$$.
See the step by step solution

## Step 1: Write down the given matrix

First, let's write down the given matrix A: $A=\begin{pmatrix} 0 & 1 & 5 \\ 3 & -6 & 9 \\ 2 & 6 & 1 \end{pmatrix}$

## Step 2: Choose a row or column to expand along

Next, choose a row or column to expand the determinant along. In this case, we'll choose the first row since it contains a 0, so it will simplify our calculations. $\det(A) = 0 \cdot \det(M_{11}) - 1 \cdot \det(M_{12}) + 5 \cdot \det(M_{13})$

## Step 3: Calculate Minor matrices

Now, we have to find the 2x2 minor matrices (Mij) where i=1 and j= 1,2,3. $M_{11} = \begin{pmatrix} -6 & 9 \\ 6 & 1 \end{pmatrix}, \ M_{12} = \begin{pmatrix} 3 & 9 \\ 2 & 1 \end{pmatrix}, \ M_{13} = \begin{pmatrix} 3 & -6 \\ 2 & 6 \end{pmatrix}$

## Step 4: Find the determinant of each minor

Next, we'll find the determinant of each minor matrix: $\det(M_{11}) = (-6)(1) - (9)(6) = -60 \\ \det(M_{12}) = (3)(1) - (9)(2) = -15 \\ \det(M_{13}) = (3)(6) - (-6)(2) = 24$

## Step 5: Substitute the values into the determinant formula

Substitute the calculated determinant values of minor matrices back into the previous formula: $\det(A) = 0 \cdot (-60) - 1 \cdot (-15) + 5 \cdot (24) \\$

## Step 6: Calculate the determinant of matrix A

Finally, we'll calculate the determinant of matrix A: $\det(A) = 0 + 15 + 120 = 135 \\$ The determinant of matrix A is 135.

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