7.3 Transpose, column operations, and Cramer's rule

The previous note studies determinants from the row-operation viewpoint. This note turns the picture sideways.

Transpose lets us move information between rows and columns. Once that symmetry is in place, column expansion and column operations become just as legitimate as their row versions. The chapter then closes with two classical formulas: adjoints and Cramer's rule.

Transpose does not change determinant

Theorem

Determinant of a transpose

For every square matrix $A$ ,

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

This theorem is conceptually important. A determinant is defined by expanding along rows, but it does not secretly prefer rows over columns. Transpose swaps the two viewpoints without changing the final scalar.

Theorem

Cofactor expansion along any column

For any fixed column j of a square matrix $A=[a_{ij}]$ ,

\det(A)=a_{1j}A_{1j}+a_{2j}A_{2j}+\cdots+a_{nj}A_{nj}.

Equivalently,

\det(A)=\sum_{i=1}^n (-1)^{i+j}a_{ij}M_{ij}.

So you may expand along a row or along a column. The right choice is whichever one produces the cleanest minors.

Worked example

A column expansion with many zeros

Let

A= \begin{bmatrix} 1&7&0\\ 6&9&8\\ 0&1&5 \end{bmatrix}.

The third column has one zero, so expand along that column:

\det(A)= 0\cdot A_{13} +8A_{23} +5A_{33}.

That gives

\det(A) =-8 \begin{vmatrix} 1&7\\ 0&1 \end{vmatrix} +5 \begin{vmatrix} 1&7\\ 6&9 \end{vmatrix} =-8(1)+5(9-42)=-173.

The same answer would come from row expansion, but the chosen column makes the arithmetic shorter.

Column operations obey the same pattern

Because transpose preserves determinant, every row-operation rule has a column version.

Theorem

How column operations change determinant

Let $B$ be obtained from a square matrix $A$ by one elementary column operation.

Swapping two columns multiplies the determinant by $-1$ .
Multiplying one column by $\beta$ multiplies the determinant by $\beta$ .
Replacing one column with itself plus a multiple of another column leaves the determinant unchanged.

Worked example

Use column operations to create zeros

Consider

C= \begin{bmatrix} 3&2&-1\\ 4&1&6\\ -3&-1&2 \end{bmatrix}.

Apply the column operations $-C_2+C_1$ and $C_3+C_1$ :

\begin{bmatrix} 3&2&-1\\ 4&1&6\\ -3&-1&2 \end{bmatrix} \longrightarrow \begin{bmatrix} 1&2&0\\ 3&1&9\\ -2&-1&-1 \end{bmatrix}.

These are column-addition operations, so the determinant is unchanged. Now expand along the third column:

\det(C)= 9 \begin{vmatrix} 1&2\\ -2&-1 \end{vmatrix} -1 \begin{vmatrix} 1&2\\ 3&1 \end{vmatrix} =9(3)-(-5)=22.

The point is not that column operations are always superior. The point is that you may use whichever direction creates more zeros with less bookkeeping.

Adjoint matrices package all cofactors at once

The cofactor of one entry helps in one expansion. The adjoint matrix collects all cofactors into one object.

Definition

Adjoint matrix

For an $n\times n$ matrix $A=[a_{ij}]$ , first form the cofactor matrix

\begin{bmatrix} A_{11}&A_{12}&\cdots&A_{1n}\\ A_{21}&A_{22}&\cdots&A_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ A_{n1}&A_{n2}&\cdots&A_{nn} \end{bmatrix}.

The adjoint matrix is the transpose of that cofactor matrix:

\operatorname{adj}(A)= \begin{bmatrix} A_{11}&A_{21}&\cdots&A_{n1}\\ A_{12}&A_{22}&\cdots&A_{n2}\\ \vdots&\vdots&\ddots&\vdots\\ A_{1n}&A_{2n}&\cdots&A_{nn} \end{bmatrix}.

Theorem

Adjoint identity and inverse formula

For every square matrix $A$ ,

A\,\operatorname{adj}(A)=\det(A)I.

If $\det(A)\neq0$ , then $A$ is invertible and

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

This formula is conceptually clean, but in numerical work it is usually less efficient than row reduction. Its value is that it exposes the algebraic structure of inverse matrices.

Worked example

Recover the 2×2 inverse formula from the adjoint

Let

A= \begin{bmatrix} a&b\\ c&d \end{bmatrix}, \qquad \det(A)=ad-bc\neq0.

The cofactor matrix is

\begin{bmatrix} d&-c\\ -b&a \end{bmatrix},

\operatorname{adj}(A)= \begin{bmatrix} d&-b\\ -c&a \end{bmatrix}.

Therefore

A^{-1}= \frac{1}{ad-bc} \begin{bmatrix} d&-b\\ -c&a \end{bmatrix}.

The familiar inverse formula is really the adjoint identity written in the $2\times2$ case.

Cramer's rule solves one coordinate at a time

Theorem

Cramer's rule

Let $A$ be an invertible $n\times n$ matrix and let $b\in\mathbb{R}^n$ . For each j, let $M_j$ be the matrix obtained by replacing the jth column of $A$ with b.

If x is the unique solution of

Ax=b,

then

x_j=\frac{\det(M_j)}{\det(A)}.

Cramer's rule is beautiful because each coordinate is isolated by one determinant ratio. It is not the fastest method for large systems, but it is a clean theoretical formula for square invertible systems.

Worked example

Solve a 2×2 system with Cramer's rule

Solve

\begin{cases} 2x_1+x_2=5,\\ x_1+3x_2=7. \end{cases}

Write

A= \begin{bmatrix} 2&1\\ 1&3 \end{bmatrix}, \qquad b= \begin{bmatrix} 5\\ 7 \end{bmatrix}.

First compute

\det(A)=2\cdot3-1\cdot1=5.

Replace the first column by b:

M_1= \begin{bmatrix} 5&1\\ 7&3 \end{bmatrix}, \qquad \det(M_1)=15-7=8.

Replace the second column by b:

M_2= \begin{bmatrix} 2&5\\ 1&7 \end{bmatrix}, \qquad \det(M_2)=14-5=9.

x_1=\frac{8}{5}, \qquad x_2=\frac{9}{5}.

Common mistake

Cramer's rule is not a universal system solver

Cramer's rule requires a square coefficient matrix and a nonzero determinant. If the system is rectangular or singular, the rule is not available. Even when it applies, it is usually less efficient than Gaussian elimination for large systems.

Quick check

Does transpose change the determinant of a square matrix?

State the theorem directly.

Solution

Answer

Quick check

What column operation leaves the determinant unchanged?

Think of the column analogue of a type-III row operation.

Solution

Answer

Quick check

When may Cramer's rule be used?

Identify the structural hypotheses on A.

Solution

Answer

Exercises

Quick check

Use a column expansion to compute $\det\begin{bmatrix}1&0&2\\3&4&5\\0&0&6\end{bmatrix}$ .

Choose the column with the most zeros.

Solution

Guided solution

Quick check

For $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$ , write down $\operatorname{adj}(A)$ .

Compute the four cofactors first, then transpose the cofactor matrix.

Solution

Guided solution

Quick check

Use Cramer's rule to find x_1 for the system $x_1+x_2=3$ , $2x_1-x_2=0$ .

You only need $\det(A)$ and $\det(M_1)$ .

Solution

Guided solution

Return to 7.2 Row operations, products, and invertibility if the row-operation bookkeeping is still shaky.

Keep 3.2 Transpose and special matrices nearby, because the transpose theorem here extends that earlier structural chapter.

For system language, connect this note back to 1.1 Equations and solution sets and 2.3 Gaussian elimination and RREF.

7.3 Transpose, column operations, and Cramer's rule

MATH1030: Linear algebra I

Transpose does not change determinant

Determinant of a transpose

Cofactor expansion along any column

A column expansion with many zeros

Column operations obey the same pattern

How column operations change determinant

Use column operations to create zeros

Adjoint matrices package all cofactors at once

Adjoint matrix

Adjoint identity and inverse formula

Recover the 2×2 inverse formula from the adjoint

Cramer's rule solves one coordinate at a time

Cramer's rule

Solve a 2×2 system with Cramer's rule

Common mistake

Cramer's rule is not a universal system solver

Quick check

Does transpose change the determinant of a square matrix?

Answer

What column operation leaves the determinant unchanged?

Answer

When may Cramer's rule be used?

Answer

Exercises

Use a column expansion to compute $\det\begin{bmatrix}1&0&2\\3&4&5\\0&0&6\end{bmatrix}$ .

Guided solution

For $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$ , write down $\operatorname{adj}(A)$ .

Guided solution

Use Cramer's rule to find x_1 for the system $x_1+x_2=3$ , $2x_1-x_2=0$ .

Guided solution

Section mastery checkpoint

Fill in the blank: for an invertible square system Ax=b, Cramer's rule says x_j = ____.

Prerequisites

Key terms in this unit

More notes in this series

7.3 Transpose, column operations, and Cramer's rule

MATH1030: Linear algebra I

Transpose does not change determinant

Determinant of a transpose

Cofactor expansion along any column

A column expansion with many zeros

Column operations obey the same pattern

How column operations change determinant

Use column operations to create zeros

Adjoint matrices package all cofactors at once

Adjoint matrix

Adjoint identity and inverse formula

Recover the 2×2 inverse formula from the adjoint

Cramer's rule solves one coordinate at a time

Cramer's rule

Solve a 2×2 system with Cramer's rule

Common mistake

Cramer's rule is not a universal system solver

Quick check

Does transpose change the determinant of a square matrix?

Answer

What column operation leaves the determinant unchanged?

Answer

When may Cramer's rule be used?

Answer

Exercises

Use a column expansion to compute det⁡[102345006]\det\begin{bmatrix}1&0&2\\3&4&5\\0&0&6\end{bmatrix}det​130​040​256​​.

Guided solution

For A=[1234]A=\begin{bmatrix}1&2\\3&4\end{bmatrix}A=[13​24​], write down adj⁡(A)\operatorname{adj}(A)adj(A).

Guided solution

Use Cramer's rule to find x_1 for the system x1+x2=3x_1+x_2=3x1​+x2​=3, 2x1−x2=02x_1-x_2=02x1​−x2​=0.

Guided solution

Related notes

Section mastery checkpoint

Prerequisites

Key terms in this unit

More notes in this series

Use a column expansion to compute $\det\begin{bmatrix}1&0&2\\3&4&5\\0&0&6\end{bmatrix}$ .

For $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$ , write down $\operatorname{adj}(A)$ .

Use Cramer's rule to find x_1 for the system $x_1+x_2=3$ , $2x_1-x_2=0$ .