Minor and cofactor expansions

The minor of the $(i,j)$th entry of a matrix $\bfA$ is the determinant of the submatrix obtained by removing the $i$th row and the $j$th column of $\bfA$.

To compute a determinant by the a minor and cofactor expansion:

Choose a row or column.

For each entry in that row or column, form the minor by removing its entire row and column
Form the sum of each entry with the its minor. Make sure the signs of each term follow a checkerboard pattern

Practice problems

Let $\bfa = \langle a_1, a_2, a_3 \rangle, \bfb = \langle b_1, b_2, b_3\rangle, \bfc = \langle c_1, c_2, c_3 \rangle.$ Show that the triple product of the vectors equals the determinant of a matrix whose rows is given by the vectors:$$\bfa \cdot (\bfb \times \bfc) = \begin{vmatrix}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$$
Solution
For a $2 \times 2$ matrix, if one row is a multiple of the other, the determinant is zero. Use this fact and the method of minors and cofactors to show that the determinant of a $3 \times 3$ matrix is zero if one row is a multiple of another.
Solution
Compute the determinant $$\text{det } \begin{pmatrix} 1 & 5 & 0 \\ 2 & 1 & 0 \\ 1 & 0 & 3 \end{pmatrix}$$ by minors and cofactors along the second column.
Solution
Compute the determinant $$\text{det } \begin{pmatrix} 1 & 5 & 0 \\ 2 & 1 & 0 \\ 1 & 0 & 3 \end{pmatrix}$$ by minors and cofactors along the first row.
Solution
Find $\bfA^{-1}$ and use it to solve $\bfA \bfx = \bfb$, where $$\bfA = \begin{pmatrix}1 & -1& 1 \\ 1 & 1 & 1\\1 & 2 & 4 \end{pmatrix}, \quad \bfb = \begin{pmatrix} 1\\3\\2 \end{pmatrix}$$
Solution

Minor and cofactor expansions

Related topics

Practice problems