§3 Determinants, Adjoints, and Cramer’s Rule

1. The Big Idea

A determinant is a single number attached to a square matrix.

That number carries a lot of information. It can tell us whether a matrix is invertible, whether a square system has a unique solution, and whether rows or columns are secretly dependent on each other.

Only square matrices have determinants.

The most important test is:

For a square system, $\det(A)\neq 0$ means the system has exactly one solution. If $\det(A)=0$, something collapses: the rows or columns do not contain enough independent information to undo the matrix.

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2. Determinant of a $2\times 2$ Matrix

For

$$A= \begin{bmatrix} a&b\\ c&d \end{bmatrix}$$

the determinant is

$$\det(A)=ad-bc$$

You may also see determinant notation written as

$$|A|=ad-bc$$

Example

Let

$$A= \begin{bmatrix} 3&2\\ 1&-3 \end{bmatrix}$$

Then

$$\det(A)=3(-3)-2(1)$$

so

$$\det(A)=-9-2=-11$$

Since $\det(A)\neq 0$, the matrix is invertible.

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3. What the Determinant Tells You

The determinant is often used as a quick decision tool.

For a square matrix $A$:

So when you see a determinant problem, ask first: is the goal to compute a number, or is the goal to decide invertibility/uniqueness?

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4. Minors and Cofactors

For larger matrices, determinants are built from smaller determinants.

The minor $M_{ij}$ is found by deleting row $i$ and column $j$, then taking the determinant of what remains.

The cofactor $C_{ij}$ is the minor with a sign attached:

$$C_{ij}=(-1)^{i+j}M_{ij}$$

For a $3\times 3$ matrix, the sign pattern is

$$\begin{bmatrix} +&-&+\\ -&+&-\\ +&-&+ \end{bmatrix}$$

That means cofactors alternate signs like a checkerboard.

How to Find a Cofactor

To find $C_{ij}$:

1. Delete row $i$.
2. Delete column $j$.
3. Compute the determinant of the smaller matrix. That is $M_{ij}$.
4. Apply the sign $(-1)^{i+j}$.

The minor is the smaller determinant. The cofactor is the signed minor.

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5. Cofactor Expansion

Cofactor expansion is a general method for computing determinants.

The idea is to choose a row or column, multiply each entry by its cofactor, and add the results.

Expansion along row $i$:

$$\det(A)=a_{i1}C_{i1}+a_{i2}C_{i2}+\cdots+a_{in}C_{in}$$

Expansion along column $j$:

$$\det(A)=a_{1j}C_{1j}+a_{2j}C_{2j}+\cdots+a_{nj}C_{nj}$$

Practical Tip

Choose the row or column with the most zeros.

If an entry is zero, then its whole term becomes zero, so you do not need to compute that cofactor. This can save a lot of work.

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6. Shortcut for $3\times 3$ Determinants

For

$$A= \begin{bmatrix} a&b&c\\ d&e&f\\ g&h&i \end{bmatrix}$$

the $3\times 3$ shortcut gives

$$\det(A)=aei+bfg+cdh-ceg-bdi-afh$$

Your notes also use the “copy the first two columns” diagonal method. That method is another way to remember the same formula.

Important warning: this shortcut works only for $3\times 3$ matrices. Do not try to extend the diagonal-copy trick to $4\times 4$ matrices.

For larger matrices, use cofactor expansion or row-reduction methods.

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7. Determinants and Row Operations

Row operations are useful for determinants, but you must track how each operation changes the determinant.

In symbols:

$$\text{row swap}\quad\Longrightarrow\quad \det(B)=-\det(A)$$ $$R_i\to kR_i\quad\Longrightarrow\quad \det(B)=k\det(A)$$ $$R_i\to R_i+kR_j\quad\Longrightarrow\quad \det(B)=\det(A)$$

These rules are especially useful when reducing a matrix to triangular form. Once the matrix is triangular, the determinant is much easier to compute.

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8. Fast Determinant Facts

Some determinant results can be recognized immediately.

Triangular Matrices

If $A$ is triangular, meaning all entries above or below the main diagonal are zero, then

$$\det(A)=a_{11}a_{22}\cdots a_{nn}$$

So for triangular matrices, just multiply the diagonal entries.

Zero Row or Column

If $A$ has a row or column of zeros, then

$$\det(A)=0$$

There is not enough information in the rows or columns for the matrix to be invertible.

Repeated Rows or Columns

If $A$ has two equal rows or two equal columns, then

$$\det(A)=0$$

Multiple Rows or Columns

If one row is a multiple of another row, or one column is a multiple of another column, then

$$\det(A)=0$$

This is a sign of linear dependence.

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9. Determinants and Matrix Operations

For square matrices $A$ and $B$ of the same size:

$$\det(A^T)=\det(A)$$ $$\det(AB)=\det(A)\det(B)$$

If $A$ is invertible:

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

For an $n\times n$ matrix:

$$\det(kA)=k^n\det(A)$$

The last rule is easy to misuse. Multiplying a whole $n\times n$ matrix by $k$ multiplies every row by $k$, so the determinant gets $n$ factors of $k$.

Important Warning

Determinants do not distribute over addition:

$$\det(A+B)\neq \det(A)+\det(B)$$

in general.

They also do not distribute over subtraction:

$$\det(A-B)\neq \det(A)-\det(B)$$

in general.

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10. The Adjoint Matrix

The adjoint is built from cofactors.

First, form the cofactor matrix:

$$C= \begin{bmatrix} C_{11}&C_{12}&\cdots\\ C_{21}&C_{22}&\cdots\\ \vdots&\vdots&\ddots \end{bmatrix}$$

Then transpose it.

The adjoint of $A$, written $\operatorname{adj}(A)$, is

$$\operatorname{adj}(A)=C^T$$

So the process is:

1. Find every cofactor.
2. Put the cofactors into a matrix.
3. Transpose that matrix.

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11. Inverse Using the Adjoint

If $A$ is invertible, then

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

This formula works for any square invertible matrix.

For large matrices, row-reduction is usually easier. Still, this formula is important because it connects three major ideas:

• determinants
• cofactors/adjoints
• inverses

It also explains why $\det(A)\neq 0$ is required. If $\det(A)=0$, then the formula would require division by zero.

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12. Useful Adjoint Properties

For an invertible $n\times n$ matrix:

$$\operatorname{adj}(A)=\det(A)A^{-1}$$ $$\det(\operatorname{adj}(A))=\det(A)^{n-1}$$ $$\operatorname{adj}(AB)=\operatorname{adj}(B)\operatorname{adj}(A)$$ $$\operatorname{adj}(A^{-1})=\frac{1}{\det(A)}A$$ $$\operatorname{adj}(kA)=k^{n-1}\operatorname{adj}(A)$$

These are useful for formula-based determinant problems. Notice that the product rule reverses order, just like inverse and transpose product rules do.

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13. Cramer’s Rule

Cramer’s Rule solves square systems using determinants.

Suppose a system is written as

$$A\vec{x}=\vec{b}$$

and

$$\det(A)\neq 0$$

Then the system has a unique solution.

For variables $x_1,x_2,\dots,x_n$,

$$x_i=\frac{\det(A_i)}{\det(A)}$$

where $A_i$ is the matrix formed by replacing column $i$ of $A$ with the constant vector $\vec{b}$.

In words:

• replace the first column to find $x_1$
• replace the second column to find $x_2$
• keep going one column at a time

Cramer’s Rule is most practical for small systems, especially $2\times 2$ or sometimes $3\times 3$ systems.

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14. Worked Example: Cramer’s Rule

Solve the system

$$\begin{cases} 3x+2y=7\\ x-3y=-5 \end{cases}$$

The coefficient matrix is

$$A= \begin{bmatrix} 3&2\\ 1&-3 \end{bmatrix}$$

and the constant vector is

$$\vec{b}= \begin{bmatrix} 7\\ -5 \end{bmatrix}$$

First compute the determinant of $A$:

$$\det(A)=3(-3)-2(1)=-11$$

Since $\det(A)\neq 0$, Cramer’s Rule applies.

Find $x$

Replace the $x$-column with $\vec{b}$:

$$A_x= \begin{bmatrix} 7&2\\ -5&-3 \end{bmatrix}$$

Then

$$\det(A_x)=7(-3)-2(-5)=-21+10=-11$$

So

$$x=\frac{\det(A_x)}{\det(A)}=\frac{-11}{-11}=1$$

Find $y$

Replace the $y$-column with $\vec{b}$:

$$A_y= \begin{bmatrix} 3&7\\ 1&-5 \end{bmatrix}$$

Then

$$\det(A_y)=3(-5)-7(1)=-15-7=-22$$

So

$$y=\frac{\det(A_y)}{\det(A)}=\frac{-22}{-11}=2$$

Therefore,

$$(x,y)=(1,2)$$

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15. Problem-Solving Routine

When working with determinants, use this checklist:

1. Check that the matrix is square. If it is not square, it has no determinant.
2. For $2\times 2$, use $ad-bc$. This is the fastest method.
3. For $3\times 3$, use the shortcut or cofactor expansion. The shortcut works only for $3\times 3$.
4. For larger matrices, look for zeros. Cofactor expansion is easier along rows or columns with many zeros.
5. If using row operations, track determinant changes. Swaps, scaling, and row replacement affect determinants differently.
6. For invertibility, only check whether the determinant is zero. Nonzero means invertible; zero means not invertible.
7. For Cramer’s Rule, replace one column at a time. Each replaced-column determinant gives one variable.

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16. Key Takeaways

• Determinants are only defined for square matrices.
• $\det(A)\neq 0$ means $A$ is invertible.
• $\det(A)=0$ means $A$ is not invertible.
• Cofactors are signed minors.
• Cofactor expansion breaks a determinant into smaller determinants.
• Row swaps change determinant sign.
• Scaling one row scales the determinant.
• Adding a multiple of one row to another does not change the determinant.
• The adjoint is the transpose of the cofactor matrix.
• Cramer’s Rule solves square systems using determinant ratios.

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Mini-Self-Check

1. What does $\det(A)=0$ tell you about a square matrix $A$?

2. Which row operation does not change the determinant?

3. Why does Cramer’s Rule require $\det(A)\neq 0$?

4. How do you build the adjoint matrix?