Consider the generic \(2\times 2\) matrix \(A\) and the “mixed up” version \(A_{\mathrm{mix}}\text{:}\)
\begin{align*}
A \amp = \begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix},
\amp
A_{\mathrm{mix}} \amp = \left[\begin{array}{rr} d \amp -c \\ -b \amp a \end{array}\right].
\end{align*}
(a)
Compute \(A\utrans{A}_\mathrm{mix}\text{.}\) Then fill in the blank.
Recall that if the product of two square matrices is equal to \(I\text{,}\) then those matrices must be inverses of each other (Proposition 6.5.4 and Proposition 6.5.6). With this knowledge, compare equation (✶✶) with Proposition 5.5.4.
(d)
What needs to be true about \(a,b,c,d\) for the algebra in Task b to be valid? Why?
The goal of this and the next two discovery guides (along with the corresponding chapters) is to develop something similar to the results of the first discovery activity above for larger square matrices. First, we will start by extending the \(2\times 2\) formula \(ad-bc\text{.}\) This formula determines whether a \(2\times 2\) matrix is invertible or not, so we call it the determinant of the matrix.
We will actually start back at \(1\times 1\) matrices, and build up from there.
Discovery8.2.
Consider the generic \(1\times 1\) matrix \(A = \begin{bmatrix}a\end{bmatrix}\text{.}\)
(a)
The inverse of \(A = \begin{bmatrix}a\end{bmatrix}\) is \(\inv{A} = \begin{bmatrix}\fillinmath{XXXXX}\end{bmatrix}\text{,}\) but this only works if .
(b)
So before attempting to compute \(\inv{A}\text{,}\) we can determine whether this attempt will be successful by looking at the matrix \(A = \begin{bmatrix} a \end{bmatrix}\) and considering the single number .
(Make sure your response is always a number!)
To build up to larger matrices, we need to take it step-by-step.
Discovery8.3.
For an \(n \times n\) matrix with \(n \gt 1\text{,}\) the \(\nth[(i,j)]\) minor (denoted \(M_{ij}\)) is the determinant of the smaller submatrix obtained by removing the row and column that contain the \(\nth[(i,j)]\) entry.
Since you know how to compute \(1\times 1\) determinants, you can now compute all four minors (\(M_{11},M_{12},M_{21},M_{22}\)) of the matrix
The \(\nth[(i,j)]\) cofactor of a matrix (denoted \(C_{ij}\)) is defined to be the \(\nth[(i,j)]\) minor, except that we multiply it by \(-1\) when \(i+j\) is odd. That is, \(C_{ij} = (-1)^{i+j} M_{ij}\text{.}\) Compute all four cofactors (\(C_{11},C_{12},C_{21},C_{22}\)) for the matrix from Discovery 8.3. (You’ve already computed the minors, now you just need to make some of them negative.)
Discovery8.5.
We now initially define the determinant of a matrix to be a combination of entries and cofactors along the first row. To compute the determinant, multiply each entry in the first row by its own cofactor, and then add all these together. For a \(2\times 2\) matrix, the formula is
\begin{equation*}
\det A = a_{11}C_{11} + a_{12}C_{12}.
\end{equation*}
Use this formula to compute the determinant of the matrix from Discovery 8.3.
Discovery8.6.
Use \(\det A = a_{11}C_{11} + a_{12}C_{12}\) to compute the determinant of the generic \(2\times 2\) matrix
\begin{equation*}
\begin{bmatrix}a \amp b \\ c \amp d\end{bmatrix}.
\end{equation*}
Surprised?
Discovery8.7.
Compute the determinant of the \(3\times 3\) matrix
Use the same sort of “cofactor expansion along the first row” as before; that is, “entry times cofactor plus entry times cofactor plus entry times cofactor …” along the first row, except now your cofactor calculations will involve \(2\times 2\) determinants.
Discovery8.8.
For this activity, use the same matrix as Discovery 8.7.
(a)
Try computing a cofactor expansion along a different row.
(b)
Now try along a column.
What did you find in these calculations? Make a conjecture about cofactor expansions along different rows or columns in a matrix in general.
Discovery8.9.
Recall the cofactor formula: \(C_{ij} = (-1)^{i+j} M_{ij}\text{.}\) The \((-1)^{i+j}\) part will be \(1\) when \(i+j\) is even and \(-1\) when \(i+j\) is odd. In a \(2\times 2\) matrix this makes a pattern: \(\left[\begin{smallmatrix}+ \amp -\\- \amp +\end{smallmatrix}\right]\text{.}\)
Make similar matrices of \(+/-\) for the patterns of cofactor signs in \(3\times 3\) and \(4\times 4\) matrices.
Discovery8.10.
(a)
Using your finding from Discovery 8.8 as appropriate, come up with simple formulas for the determinant of diagonal matrices, upper triangular matrices, and lower triangular matrices.
Hint.
In these special matrices, there are some rows/columns that are easier to use in a cofactor expansion than others.
(b)
What is \(\det \zerovec\text{?}\) … \(\det I\text{?}\) Are the answers the same for every size of zero/identity matrix?
