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
#1:
Instructions: find the matrix of cofactors.
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cc}3 & -1\\ 5 & 7\end{array}\right]$$
$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cc}-2 & -1\\ 5 & 0\end{array}\right]$$
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#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.
$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cc}3 & -1\\ 2 & -7\end{array}\right]$$
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#3:
Instructions: find the determinant.
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cc}-4 & 6\\ -1 & 0\end{array}\right]$$
$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cc}3 & 1\\ 5 & -10\end{array}\right]$$
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#4:
Instructions: find the determinant.
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{ccc}-1 & 0 & 5\\ 1 & 2 & -3 \\6 & -2 & -10\end{array}\right]$$
$$b)\hspace{.2em}$$ $$\left[ \begin{array}{ccc}1 & 1 & -5\\ 2 & 5 & 7 \\0 & 3 & 0\end{array}\right]$$
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#5:
Instructions: find the determinant.
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{ccc}-2 & -7 & 8\\ -4 & 1 & 0 \\0 & -2 & 9\end{array}\right]$$
$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cccc}1 & 9 & 1 & -2\\ 0 & 1 & 4 & 0 \\-5 & 5 & -1 & 0 \\-4 & 7 & 6 & 3\end{array}\right]$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cc}7 & -5\\ 1 & 3\end{array}\right]$$
$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cc}0 & -5\\ 1 & -2\end{array}\right]$$
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#2:
Solutions:
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{ccc}-3 & -6 & -8\\ -1 & -2 & -7 \\-5 & -23 & -35\end{array}\right]$$
$$b)\hspace{.2em}-19$$
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#3:
Solutions:
$$a)\hspace{.2em}6$$
$$b)\hspace{.2em}-35$$
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#4:
Solutions:
$$a)\hspace{.2em}-44$$
$$b)\hspace{.2em}-51$$
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#5:
Solutions:
$$a)\hspace{.2em}-206$$
$$b)\hspace{.2em}-640$$
