### About Finding the Determinant of a Matrix:

To find the determinant of an n x n matrix (square matrix), we can use a technique known as Laplace Expansion.

Test Objectives
• Demonstrate the ability to find a matrix of cofactors
• Demonstrate the ability to find the determinant of an n x n matrix
Finding the Determinant of a Matrix Practice Test:

#1:

Instructions: find the matrix of cofactors.

$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cc}3 & -1\\ 5 & 7\end{array}\right]$$

$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cc}-2 & -1\\ 5 & 0\end{array}\right]$$

#2:

Instructions: find the matrix of cofactors.

$$a)\hspace{.2em}$$ $$\left[ \begin{array}{ccc}7 & 0 & -1\\ 2 & -5 & 3 \\ -2 & 1 & 0\end{array}\right]$$

Instructions: find the determinant.

$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cc}3 & -1\\ 2 & -7\end{array}\right]$$

#3:

Instructions: find the determinant.

$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cc}-4 & 6\\ -1 & 0\end{array}\right]$$

$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cc}3 & 1\\ 5 & -10\end{array}\right]$$

#4:

Instructions: find the determinant.

$$a)\hspace{.2em}$$ $$\left[ \begin{array}{ccc}-1 & 0 & 5\\ 1 & 2 & -3 \\6 & -2 & -10\end{array}\right]$$

$$b)\hspace{.2em}$$ $$\left[ \begin{array}{ccc}1 & 1 & -5\\ 2 & 5 & 7 \\0 & 3 & 0\end{array}\right]$$

#5:

Instructions: find the determinant.

$$a)\hspace{.2em}$$ $$\left[ \begin{array}{ccc}-2 & -7 & 8\\ -4 & 1 & 0 \\0 & -2 & 9\end{array}\right]$$

$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cccc}1 & 9 & 1 & -2\\ 0 & 1 & 4 & 0 \\-5 & 5 & -1 & 0 \\-4 & 7 & 6 & 3\end{array}\right]$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cc}7 & -5\\ 1 & 3\end{array}\right]$$

$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cc}0 & -5\\ 1 & -2\end{array}\right]$$

#2:

Solutions:

$$a)\hspace{.2em}$$ $$\left[ \begin{array}{ccc}-3 & -6 & -8\\ -1 & -2 & -7 \\-5 & -23 & -35\end{array}\right]$$

$$b)\hspace{.2em}-19$$

#3:

Solutions:

$$a)\hspace{.2em}6$$

$$b)\hspace{.2em}-35$$

#4:

Solutions:

$$a)\hspace{.2em}-44$$

$$b)\hspace{.2em}-51$$

#5:

Solutions:

$$a)\hspace{.2em}-206$$

$$b)\hspace{.2em}-640$$