Consider the general \(3\times 3\) matrix \(A\) (below left). Each entry \(a_{ij}\) has a corresponding cofactor \(C_{ij}\text{,}\) creating a matrix of cofactors \(C_A\) (below right).
Write out the \((1,2)\) entry of the product \(A\utrans{C}_A\) as a formula in the entries of \(A\) and \(C_A\text{.}\) Does the result look familiar? What did we discover about “mixed” cofactor expansions in Discovery 9.6 and Subsubsection 9.2.1.3?
Suppose \(\det A = 0\text{.}\) If you apply some elementary row operation to \(A\text{,}\) what is the determinant of the new matrix? (Consider each of the three kinds of operations.)
Suppose \(\det A \neq 0\text{.}\) Is there any elementary row operation you can apply to \(A\) so that the new matrix has determinant \(0\text{?}\) (Consider each of the three kinds of operations.)
If \(\det A \neq 0\) and you perform a whole sequence of row operations to \(A\text{,}\) could the last matrix in the sequence have determinant \(0\text{?}\)
Recall that if a matrix is singular (that is, not invertible), then it is not possible to row reduce it to \(I\) (Theorem 6.5.2), and so its RREF must have a row of zeros. If \(\det A \neq 0\text{,}\) could \(A\) be singular?
Recall that for matrix \(A\) and elementary matrix \(E\text{,}\) the result of \(E A\) is the same as the result of performing an elementary row operation on \(A\) (namely, the operation corresponding to \(E\)). Verify the formula
\begin{gather}
\det (E A) = (\det E) (\det A) \tag{✶}
\end{gather}
for each of the three types of elementary matrices \(E\) (assuming \(A\) to be a square matrix of the same size as \(E\)).
To verify a formula, consider LHS and RHS separately, and argue that they equal the same value. Do not work with the proposed equality directly, since you don’t know it’s an equality yet.
For each type of \(E\text{,}\) on the LHS consider the product of matrices\(E A\) and how its determinant compares to \(\det A\) using the rules for how row operations affect determinant (explored in Discovery guide 9.1, and recalled for you at the top of this activity section). For this, think of \(\det A = \det (\text{old})\) and \(\det (E A) = \det (\text{new})\text{.}\) Then, on the RHS, consider the value of \(\det E\) and the corresponding product of numbers\((\det E) (\det A)\text{.}\)
Recall that every invertible matrix can be expressed as a product of elementary matrices (Theorem 6.5.2). For now, suppose that \(M\) (which we have assumed invertible) can be expressed as a product of three elementary matrices, say \(M = E_1 E_2 E_3\text{.}\) Use formula (✶) to verify that
Do you think the calculations in this activity would work out similarly no matter how many \(E_i\)’s are required to express \(M\) as a product of elementary matrices?
Remember that a fraction does not make sense for matrices. However, \(\det A\) is just a number, so you can do all the normal algebra you would like with it!
In this discovery activity, we extend formula (✶✶) to also be valid in case that \(M\) is singular (assuming \(M\) and \(N\) to be square matrices of the same size).
