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Determinants and Inverses
Determinants and Inverses
The determinant helps decide whether a square matrix is invertible.
2x2 Determinant
For $$A=\begin{bmatrix}a&b\c&d\end{bmatrix},$$ $$\det(A)=ad-bc.$$
Example 1
Find the determinant of $$\begin{bmatrix}2&3\1&4\end{bmatrix}.$$
$2(4)-3(1)=5$.
Invertibility Test
A square matrix is invertible if and only if its determinant is nonzero.
2x2 Inverse
If $$A=\begin{bmatrix}a&b\c&d\end{bmatrix}$$ and $ad-bc \ne 0$, then $$A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\-c&a\end{bmatrix}.$$
Example 2
Find the inverse of $$\begin{bmatrix}1&2\3&4\end{bmatrix}.$$
Determinant $=-2$, so $$A^{-1}=\frac{1}{-2}\begin{bmatrix}4&-2\-3&1\end{bmatrix}. $$
Practice Problems
1. Find $$\det\begin{bmatrix}5&1\2&3\end{bmatrix}.$$
2. Is $$\begin{bmatrix}2&4\1&2\end{bmatrix}$$ invertible?
3. State the condition for invertibility.
4. What is the determinant of the identity matrix?
5. Why can a zero determinant be a problem for solving systems?
Show Answer Key
1. $13$
2. No; determinant is $0$
3. Determinant not equal to zero
4. $1$
5. The coefficient matrix is singular, so a unique solution does not exist.