3.2: Properties of Determinants

Learning Objectives

• Verify and apply the following properties:
  • \(\det(AB)=\det(A)\det(B)\) (determinant of a product)
  • \(\det(A^T)=\det(A)\) (determinant of a transpose)
  • \(\det(A^{-1})=\dfrac{1}{\det(A)}\) when invertible
  • \(\det(kA)=k^n\det(A)\) for \(n\times n\) matrices
• Demonstrate the effects that elementary row operations have on determinants.

Properties of Determinants (Core)

Before we explore how row operations affect determinants, it helps to gather the most fundamental properties first. These properties connect determinants to matrix multiplication, inverses, transposes, and scaling by constants. Together they form the “toolkit” you will repeatedly use in computations and proofs.

Theorem \(\PageIndex{5}\): Determinant of a Product

If \(A,B\) are \(n\times n\) matrices, then \(\det(AB)=\det(A)\det(B).\)

Proof (click to expand)

If \(A\) is an elementary matrix, multiplying by \(A\) on the left performs a row operation and the formula follows. For a general \(A\), write \(A=E_1\cdots E_mC\) with \(C\) in reduced row echelon form. If \(C=I\), repeated application proves the result. If \(C\) has a zero row, both \(\det(A)\) and \(\det(AB)\) are zero. Thus in all cases, \(\det(AB)=\det(A)\det(B).\)

This property is especially important: it tells us determinants respect matrix multiplication in the same way as absolute values respect multiplication of numbers.

Theorem \(\PageIndex{6}\): Determinant of the Transpose

\(\det(A^T)=\det(A)\).

Proof (click to expand)

For elementary matrices, \(\det(E)=\det(E^T)\). Writing \(A=E_1\cdots E_mC\), we see \(A^T=C^TE_m^T\cdots E_1^T\). Since \(\det(E_j)=\det(E_j^T)\) and \(\det(C)=\det(C^T)\), it follows \(\det(A^T)=\det(A).\)

Notice that this means determinants do not depend on whether we view a matrix by rows or by columns. This symmetry will also justify why column operations behave like row operations.

Theorem \(\PageIndex{7}\): Determinant of the Inverse

If \(A\) is invertible, then \(\det(A^{-1})=\dfrac{1}{\det(A)}\).

Proof (click to expand)

Since \(AA^{-1}=I\), \(\det(A)\det(A^{-1})=\det(I)=1.\) Hence \(\det(A^{-1})=1/\det(A).\)

This connects determinants to invertibility: a matrix is invertible exactly when its determinant is nonzero. Thus, determinants provide a quick “test for invertibility.”

Theorem \(\PageIndex{3}\): Scalar Multiplication

If \(B=kA\), then \(\det(B)=k^n\det(A).\)

Proof (click to expand)

Multiplying each row of \(A\) by \(k\) multiplies the determinant by \(k\). Doing this for all \(n\) rows multiplies by \(k^n\), so \(\det(kA)=k^n\det(A).\)

Together, these four properties give us a powerful set of shortcuts when evaluating determinants or analyzing matrices. Next we turn to row operations, where we see how basic manipulations affect determinant values.

Row Operations and Determinants

In Chapter 1, row operations were used to simplify matrices and solve systems. Now we revisit them with a new perspective: every row operation has a predictable effect on the determinant. Understanding this allows us to compute determinants of large matrices more efficiently.

Theorem (Row Operations)

1. Swapping two rows multiplies the determinant by \(-1\).
2. Multiplying a row by \(k\) multiplies the determinant by \(k\).
3. Adding a multiple of one row to another leaves the determinant unchanged.

Proof (click to expand)

Each case is checked directly for \(2\times 2\) matrices, then extended by induction. For example, if two rows are identical, interchanging them shows \(\det A=-\det A\), so \(\det A=0\). Multiplying a row scales the determinant by that factor. Adding a multiple of a row to another preserves cofactors, so the determinant does not change.

These rules mean that when simplifying a matrix using row operations, you can keep track of how the determinant changes without having to recompute it from scratch.

Example: Switching Rows

\(A=\begin{bmatrix}1&2\\3&4\end{bmatrix}\), \(\det(A)=-2\). Swapping rows → \(\det(B)=2.\)

Example: Adding a Multiple of a Row

\(A=\begin{bmatrix}1&2\\3&4\end{bmatrix},\ B=\begin{bmatrix}1&2\\5&8\end{bmatrix}.\) Row2(B) = Row2(A)+2·Row1(A). \(\det(B)=\det(A)=-2.\)

With these principles in place, we can now attempt larger matrices. The goal is to apply row operations to simplify the matrix, carefully tracking determinant changes, until the determinant can be computed by expansion or by recognizing a triangular form.

Example \(\PageIndex{10}\): Large Determinant

For \(A=\begin{bmatrix}1&2&3&4\\5&1&2&3\\4&5&4&3\\2&2&-4&5\end{bmatrix}\), row ops lead to an upper-triangular matrix with product of diagonal entries \(1485\). Adjusting for scaling, \(\det(A)=495\).

This example shows how quickly row operations reduce the amount of work compared to Laplace expansion.

Example \(\PageIndex{11}\)

For \(A=\begin{bmatrix}1&2&3&2\\1&-3&2&1\\2&1&2&5\\3&-4&1&2\end{bmatrix}\), careful row and column operations simplify it to a form where expansion gives \(\det(A)=-82\).

Row operations provide a powerful computational strategy, and combining them with the determinant properties from earlier makes it possible to handle both theory and computation with confidence.