Why Track the Whole Sequence?
Row reduction can turn a complicated square matrix into an upper triangular matrix, whose determinant is the product of its diagonal entries. But the row operations used to get there may change the determinant. To find the determinant of the original matrix, keep track of those changes as they happen. The key is to record the total multiplier from the entire sequence, rather than trying to remember the effect of each operation at the end.
Throughout this tutorial, matrices are square matrices over \(\mathbb{R}\). Write \(\mathbf{A}_0\) for the starting matrix and \(\mathbf{A}_k\) for the matrix after the \(k\)th row operation. The determinant rules for elementary row operations are:
- Swapping two rows multiplies the determinant by \(-1\).
- Multiplying a row by a nonzero scalar \(c\) multiplies the determinant by \(c\).
- Adding a multiple of one row to a different row leaves the determinant unchanged.
Thus, each operation has a factor: \(-1\) for a swap, \(c\) for scaling by \(c\), and \(1\) for adding a row multiple. The scaling factor must be nonzero because scaling by zero would replace a row by a zero row and would not be an elementary row operation.
The Tracking Rule
The total multiplier records the direction in which the operations were applied: it tells how the final determinant relates to the original determinant. To recover the original determinant, divide the final determinant by the total multiplier. The rule works whether the final matrix is triangular or not; triangular form is useful because it makes the final determinant quick to compute.
Proof. For each \(k\) from \(1\) to \(m\), the determinant rule for the \(k\)th row operation gives \(\det\mathbf{A}_k=f_k\det\mathbf{A}_{k-1}\). Substituting the first relation into the second gives \(\det\mathbf{A}_2=f_2f_1\det\mathbf{A}_0\). Continuing this substitution through all \(m\) operations yields \(\det\mathbf{A}_m=f_mf_{m-1}\cdots f_1\det\mathbf{A}_0\). Real-number multiplication is commutative, so the factors can be written in the stated order. Each \(f_k\ne0\), hence their product is nonzero and division by it is valid. \(\square\)
When the final matrix is upper triangular, its determinant is the product of its diagonal entries. This formula follows from the permutation expansion of the determinant: a nonzero term must select, in row \(i\), a column numbered at least \(i\). Since each column is selected once, the only possible such selection is the diagonal. So a triangular reduction gives a direct route: multiply the diagonal entries, then divide by the tracked factor.
Set the running multiplier to \(1\), since no operations have yet been applied.
Multiply the running multiplier by \(-1\) for a swap, by \(c\) for scaling a row by \(c\), or by \(1\) for a row addition.
If the result is triangular, multiply its diagonal entries.
Divide the final determinant by the running multiplier.
Worked Sequences
Worked Example: A Swap and Two Row Additions
Track the determinant of \(\mathbf{A}=\begin{bmatrix}0&2&1\\1&1&0\\2&3&1\end{bmatrix}\) by reducing it to triangular form.
First swap rows 1 and 2. The factor is \(-1\). Then use the new row 1 to eliminate the entry below its leading entry; this row addition has factor \(1\):
Next, subtract \(\tfrac12\) of row 2 from row 3. This is also a row addition, so its factor is \(1\):
The total multiplier is \(F=(-1)(1)(1)=-1\). The final matrix is upper triangular, so its determinant is \(1\cdot2\cdot\tfrac12=1\). Therefore the original determinant is \(1/(-1)=-1\).
Check: Expanding the original determinant along its first row gives \(-2(1\cdot1-0\cdot2)+1(1\cdot3-1\cdot2)=-2+1=-1\), as tracked.
Worked Example: Including a Row Scaling
Find \(\det\mathbf{A}\) for \(\mathbf{A}=\begin{bmatrix}2&1&0\\4&3&1\\2&0&1\end{bmatrix}\), and include a row scaling in the reduction.
Apply two row additions to clear the first column below the leading entry, then add the new row 2 to row 3:
Now scale row 2 by \(\tfrac12\). The result is still upper triangular, and this operation contributes a factor of \(\tfrac12\):
The three additions each contribute \(1\), and the scaling contributes \(\tfrac12\), so \(F=\tfrac12\). The final determinant is \(2\cdot\tfrac12\cdot2=2\). Hence \(\det\mathbf{A}=2/(\tfrac12)=4\).
Check: Expansion along the first row gives \(2(3\cdot1-1\cdot0)-1(4\cdot1-1\cdot2)=6-2=4\).
Worked Example: A Zero Determinant After a Swap
Track the determinant of \(\mathbf{A}=\begin{bmatrix}1&2&1\\2&4&2\\0&1&3\end{bmatrix}\) through a reduction that produces a zero row.
Replace row 2 by row 2 minus twice row 1. This operation has factor \(1\). Then swap rows 2 and 3, which contributes \(-1\):
The final upper triangular matrix has a zero diagonal entry, so its determinant is \(1\cdot1\cdot0=0\). The total multiplier is \(F=(1)(-1)=-1\). Recovering the original determinant gives \(0/(-1)=0\).
Check: Expansion along the first row of the original matrix gives \((4\cdot3-2\cdot1)-2(2\cdot3-2\cdot0)+(2\cdot1-4\cdot0)=10-12+2=0\).
Why the Factor Can Be Reversed
The tracking rule gives both directions of the relationship. The final determinant is the original determinant multiplied by \(F\); because \(F\ne0\), the original determinant is the final determinant divided by \(F\). This also explains why a sequence of valid elementary row operations cannot turn a nonzero determinant into zero or a zero determinant into a nonzero one.
Proof. By the tracking rule, \(\det\mathbf{A}_m=F\det\mathbf{A}_0\), where \(F\ne0\). If \(\det\mathbf{A}_0=0\), then \(F\det\mathbf{A}_0=0\), so \(\det\mathbf{A}_m=0\). Conversely, if \(\det\mathbf{A}_m=0\), then \(F\det\mathbf{A}_0=0\). Since \(F\ne0\), division by \(F\) gives \(\det\mathbf{A}_0=0\). \(\square\)
Common Mistake: Multiplying in the Wrong Direction
Suppose a reduction changes \(\mathbf{A}\) to a matrix \(\mathbf{U}\) with total multiplier \(F=\tfrac12\), and \(\det\mathbf{U}=2\). A tempting but wrong calculation is \(\det\mathbf{A}=(\tfrac12)(2)=1\). That applies the forward multiplier in reverse. The forward relationship is \(\det\mathbf{U}=(\tfrac12)\det\mathbf{A}\), so solving gives \(\det\mathbf{A}=2/(\tfrac12)=4\). In the second worked example, this corrected value is confirmed by direct expansion.
A useful way to avoid this error is to write the relationship before substituting numbers: \(\det(\text{final})=F\det(\text{original})\). Then isolate the quantity being asked for. Also keep the operation ledger beside the matrices: a swap contributes \(-1\), a scaling contributes its scale factor, and a row addition contributes \(1\). Do not assign a factor of \(1\) to a scaling merely because it helps simplify the entries; its factor is the scalar actually used.
Check Your Understanding
For each sequence, identify the total multiplier and use it to recover the starting determinant from the final one.
- A row swap, followed by two row additions, produces an upper triangular matrix with diagonal entries \(2,1,3\). What is the original determinant?
- Three row additions and then scaling row 2 by \(3\) produce a final determinant of \(-6\). What was the original determinant?
- A sequence contains two row swaps and a scaling of row 1 by \(-2\). What is the total multiplier, regardless of any row additions in the sequence?
- Explain why the original and final matrices in a sequence of elementary row operations either both have zero determinant or both have nonzero determinant.
- A reduction has total multiplier \(-1\) and final determinant \(5\). Write the equation relating the original and final determinants, then find the original determinant.
