7.2 Row operations, products, and invertibility

The previous note defines determinants by cofactor expansion. That definition is correct, but it is not the way you want to compute every determinant by hand. The real power comes from combining determinants with row operations.

This note answers three questions:

how does each elementary row operation affect the determinant;
why does $\det(AB)=\det(A)\det(B)$ hold for square matrices;
why is $\det(A)\neq0$ exactly the same as saying that $A$ is invertible.

Why row operations matter

Row reduction changes a matrix into a friendlier one. The determinant is useful only if we know precisely how much that friendly matrix differs from the original.

Theorem

How the three row operations change determinant

Let $A$ be a square matrix.

If $B$ is obtained from $A$ by swapping two rows once, then

\det(B)=-\det(A).

If $B$ is obtained from $A$ by multiplying one row by a nonzero scalar $\beta$ , then

\det(B)=\beta\det(A).

If $B$ is obtained from $A$ by replacing one row with itself plus a multiple of another row, then

\det(B)=\det(A).

These are exact formulas, not rough heuristics. They are the bookkeeping rules that let elimination and determinants work together.

A row-reduction algorithm for determinants

Suppose you row-reduce a square matrix $A$ to an upper triangular matrix $U$ . Then:

track how many row swaps you used;
track every row-scaling factor you introduced;
leave row-addition steps alone, because they do not change the determinant;
read $\det(U)$ from the diagonal product.

Then undo the sign and scaling changes to recover $\det(A)$ .

Worked example

Evaluate a determinant by row reduction

Let

A= \begin{bmatrix} 1&2&3\\ 2&5&7\\ 3&8&10 \end{bmatrix}.

Apply row operations:

\begin{bmatrix} 1&2&3\\ 2&5&7\\ 3&8&10 \end{bmatrix} \xrightarrow{R_2-2R_1,\ R_3-3R_1} \begin{bmatrix} 1&2&3\\ 0&1&1\\ 0&2&1 \end{bmatrix} \xrightarrow{R_3-2R_2} \begin{bmatrix} 1&2&3\\ 0&1&1\\ 0&0&-1 \end{bmatrix}.

Only row-addition operations were used, so the determinant did not change during the reduction. The final matrix is upper triangular, so

\det(A)=1\cdot1\cdot(-1)=-1.

Worked example

A row swap changes the sign

Let

B= \begin{bmatrix} 0&1\\ 4&3 \end{bmatrix}.

Swap the two rows:

\begin{bmatrix} 0&1\\ 4&3 \end{bmatrix} \xrightarrow{R_1\leftrightarrow R_2} \begin{bmatrix} 4&3\\ 0&1 \end{bmatrix}.

The new matrix is triangular, so its determinant is 4. Because one row swap was used,

\det(B)=-4.

That agrees with the direct $2\times2$ calculation:

\det(B)=0\cdot3-1\cdot4=-4.

Multiplicativity is the structural theorem

Theorem

Determinant of a product

If $A$ and $B$ are square matrices of the same size, then

\det(AB)=\det(A)\det(B).

This theorem is deeper than it first looks. Matrix multiplication is not commutative, and usually $AB\neq BA$ . Even so,

\det(AB)=\det(A)\det(B)=\det(B)\det(A)=\det(BA).

So determinants forget some of the order-sensitive detail of matrix multiplication while preserving a very important scalar invariant.

Proof

Why row-operation matrices lead to multiplicativity

Determinant and invertibility say the same thing

Theorem

Invertibility test by determinant

For a square matrix $A$ , the following are equivalent:

$A$ is invertible.
$\det(A)\neq0$ .

This turns determinants into a full invertibility test, not merely a numerical summary.

If a row-reduced form of $A$ has a zero row, then its determinant is 0, so $A$ is not invertible. If row reduction reaches the identity matrix, then the diagonal product is nonzero, so $A$ is invertible.

Theorem

Inverse and powers

If $A$ is invertible, then

\det(A^{-1})=\frac{1}{\det(A)}.

More generally, for every positive integer m,

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

If $A$ is invertible, the same power rule remains true for negative integers as well.

Worked example

Find the parameter that makes a matrix singular

Let

A= \begin{bmatrix} 2&1&0&1\\ 0&1&1&1\\ 1&0&0&x\\ 0&2&3&1 \end{bmatrix}.

Expand along the first column:

\det(A)= 2 \begin{vmatrix} 1&1&1\\ 0&0&x\\ 2&3&1 \end{vmatrix} + \begin{vmatrix} 1&0&1\\ 1&1&1\\ 2&3&1 \end{vmatrix}.

The first $3\times3$ determinant is easiest by expanding along its second row:

\begin{vmatrix} 1&1&1\\ 0&0&x\\ 2&3&1 \end{vmatrix} =-x \begin{vmatrix} 1&1\\ 2&3 \end{vmatrix} =-x.

The second determinant equals $-1$ , so

\det(A)=2(-x)+(-1)=-2x-1.

Therefore $A$ is singular exactly when $\det(A)=0$ , that is,

x=-\frac12.

A practical checklist for hand computation

When you compute a determinant by elimination, keep the following discipline.

Use row-addition steps as freely as you like; they preserve the determinant.
Delay row scaling unless it creates a major simplification, because each scaling factor must later be divided back out.
Count row swaps carefully; each one flips the sign once.
Stop as soon as you reach an upper triangular matrix or a zero row.

This is exactly the same kind of discipline you already use in solving linear systems. The difference is that the final goal is a scalar, not a solution vector.

Common mistake

Row-equivalent matrices do not usually have the same determinant

Students often remember that row-equivalent matrices describe the same solution set for a linear system and then incorrectly conclude that they must have the same determinant. That is false.

Only row-addition steps preserve the determinant. Row swaps change its sign, and row scaling multiplies it by the scaling factor.

Quick check

If one row of A is multiplied by 5, what happens to $\det(A)$ ?

Use the row-scaling rule directly.

Solution

Answer

Quick check

If A has a zero row after row reduction, what can you conclude about $\det(A)$ ?

Think about the determinant of the echelon form first.

Solution

Answer

Quick check

Can a square matrix with determinant `0` be invertible?

Use the equivalence theorem.

Solution

Answer

Exercises

Quick check

Use row reduction to compute $\det\begin{bmatrix}1&1&0\\2&3&1\\0&1&4\end{bmatrix}$ .

Try to avoid row scaling so the bookkeeping stays short.

Solution

Guided solution

Quick check

Suppose $\det(A)=4$ and $\det(B)=-3$ for two $3\times3$ matrices. What is $\det(AB)$ ?

Use multiplicativity.

Solution

Guided solution

Quick check

If $\det(A)=6$ , what is $\det(A^{-1})$ ?

Assume A is invertible.

Solution

Guided solution

Return to 7.1 Determinants and cofactor expansion if the basic definition still feels too symbolic.

Continue with 7.3 Transpose, column operations, and Cramer's rule to see the column viewpoint and the adjoint formula.

Keep 5.1 Invertible matrices in view, because this note upgrades that chapter with a determinant criterion.

7.2 Row operations, products, and invertibility

MATH1030: Linear algebra I

Why row operations matter

How the three row operations change determinant

A row-reduction algorithm for determinants

Evaluate a determinant by row reduction

A row swap changes the sign

Multiplicativity is the structural theorem

Determinant of a product

Why row-operation matrices lead to multiplicativity

Determinant and invertibility say the same thing

Invertibility test by determinant

Inverse and powers

Find the parameter that makes a matrix singular

A practical checklist for hand computation

Common mistake

Row-equivalent matrices do not usually have the same determinant

Quick check

If one row of A is multiplied by 5, what happens to $\det(A)$ ?

Answer

If A has a zero row after row reduction, what can you conclude about $\det(A)$ ?

Answer

Can a square matrix with determinant `0` be invertible?

Answer

Exercises

Use row reduction to compute $\det\begin{bmatrix}1&1&0\\2&3&1\\0&1&4\end{bmatrix}$ .

Guided solution

Suppose $\det(A)=4$ and $\det(B)=-3$ for two $3\times3$ matrices. What is $\det(AB)$ ?

Guided solution

If $\det(A)=6$ , what is $\det(A^{-1})$ ?

Guided solution

Section mastery checkpoint

A matrix B is obtained from A by swapping two rows once. Which equation must hold?

Prerequisites

Key terms in this unit

More notes in this series

7.2 Row operations, products, and invertibility

MATH1030: Linear algebra I

Why row operations matter

How the three row operations change determinant

A row-reduction algorithm for determinants

Evaluate a determinant by row reduction

A row swap changes the sign

Multiplicativity is the structural theorem

Determinant of a product

Why row-operation matrices lead to multiplicativity

Determinant and invertibility say the same thing

Invertibility test by determinant

Inverse and powers

Find the parameter that makes a matrix singular

A practical checklist for hand computation

Common mistake

Row-equivalent matrices do not usually have the same determinant

Quick check

If one row of A is multiplied by 5, what happens to det⁡(A)\det(A)det(A)?

Answer

If A has a zero row after row reduction, what can you conclude about det⁡(A)\det(A)det(A)?

Answer

Can a square matrix with determinant 0 be invertible?

Answer

Exercises

Use row reduction to compute det⁡[110231014]\det\begin{bmatrix}1&1&0\\2&3&1\\0&1&4\end{bmatrix}det​120​131​014​​.

Guided solution

Suppose det⁡(A)=4\det(A)=4det(A)=4 and det⁡(B)=−3\det(B)=-3det(B)=−3 for two 3×33\times33×3 matrices. What is det⁡(AB)\det(AB)det(AB)?

Guided solution

If det⁡(A)=6\det(A)=6det(A)=6, what is det⁡(A−1)\det(A^{-1})det(A−1)?

Guided solution

Related notes

Section mastery checkpoint

Prerequisites

Key terms in this unit

More notes in this series

If one row of A is multiplied by 5, what happens to $\det(A)$ ?

If A has a zero row after row reduction, what can you conclude about $\det(A)$ ?

Can a square matrix with determinant `0` be invertible?

Use row reduction to compute $\det\begin{bmatrix}1&1&0\\2&3&1\\0&1&4\end{bmatrix}$ .

Suppose $\det(A)=4$ and $\det(B)=-3$ for two $3\times3$ matrices. What is $\det(AB)$ ?

If $\det(A)=6$ , what is $\det(A^{-1})$ ?