Determinants

The One Thing

A determinant is a scalar associated with a square matrix that provides a test for invertibility and supports the solution of simultaneous linear equations. Its calculation depends on minors, cofactors, row or column expansion, and determinant properties.

Definitions and Results

• Determinant: A scalar value associated with a square matrix, written as $|A|$ or $\det (A)$.
• Determinant of a $2\times 2$ matrix: For

$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],\det (A)=ad-bc.$$

• Minor: The minor $M_{ij}$ of an element $a_{ij}$ is the determinant obtained by deleting the $i$th row and $j$th column.
• Cofactor: The cofactor $C_{ij}$ is

$$C_{ij}=(-1{)}^{i+j}{M}_{ij}.$$ Cofactor signs follow the checkerboard pattern $$\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}.$$

• Expansion along a row or column: A determinant is evaluated by multiplying each element of a selected row or column by its cofactor and adding the results.
• $3\times 3$ determinant expansion: For

$$\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|,$$ expansion along the first row gives $$a(ei-fh)-b(di-fg)+c(dh-eg).$$

• Singular matrix: A square matrix whose determinant is zero. It has no inverse.
• Nonsingular matrix: A square matrix whose determinant is nonzero. It has a unique inverse.
• Adjoint: The adjoint of $A$, written $\mathrm{adj}(A)$, is the transpose of its cofactor matrix.
• Inverse of a matrix: For a nonsingular square matrix,

$$A^{-1}=\frac{\mathrm{adj}(A)}{|A|},A{A}^{-1}=A^{-1}A=I.$$ In particular, if $$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],$$ then $$A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right],$$ provided $ad-bc\ne 0$.

• Invertibility criterion: A square matrix $A$ is invertible if and only if $\det (A)\ne 0$.
• Area of a triangle: For vertices $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$,

$$\text{Area}=\frac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|.$$

• Cramer’s rule: For a system of linear equations with a nonzero coefficient determinant, each variable is found by replacing its coefficient column with the constant column and dividing the resulting determinant by the coefficient determinant.
• Determinant properties:

- Interchanging two rows or two columns changes the sign of the determinant. - If two rows or two columns are identical or proportional, the determinant is zero. - If every element of a row or column is zero, the determinant is zero. - Multiplying one row or column by a scalar $k$ multiplies the determinant by $k$. - Adding a multiple of one row or column to another does not change the determinant. - The determinant of a triangular or diagonal matrix is the product of its diagonal entries. - $\det (AB)=\det (A)\det (B)$. - $\det (A^T)=\det (A)$. - $\det (I)=1$.

• Matrix equation: If $AX=B$ and $A$ is invertible, then

$$X=A^{-1}B.$$

• Linear-system condition: If the coefficient determinant is nonzero, the system has a unique solution. If it is zero, the system may have no solution or infinitely many solutions; further comparison of determinants or equations is required.

Worked Methods

1. Evaluating a $2\times 2$ determinant

For $$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],$$ proceed as follows:

• Multiply the main diagonal entries: $ad$.
• Multiply the other diagonal entries: $bc$.
• Subtract:

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

2. Expanding a $3\times 3$ determinant

For $$\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|,$$ expand along the first row:

• Associate $a$ with the minor $ei-fh$.
• Associate $b$ with the minor $di-fg$, remembering the negative cofactor sign.
• Associate $c$ with the minor $dh-eg$.
• Combine the terms:

$$\det (A)=a(ei-fh)-b(di-fg)+c(dh-eg).$$

The same procedure can be applied along any row or column. A convenient row or column should be selected, particularly one containing zeros.

3. Using minors and cofactors

For each element $a_{ij}$:

• Delete the $i$th row and $j$th column.
• Evaluate the remaining determinant to obtain $M_{ij}$.
• Apply the sign factor:

$$C_{ij}=(-1{)}^{i+j}{M}_{ij}.$$

• Use the cofactors to expand the determinant or construct the cofactor matrix.

The signs must follow $$\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}.$$

4. Applying determinant properties

To simplify a determinant:

• Identify rows or columns that are identical, proportional, or entirely zero; the determinant is then zero.
• If useful, interchange rows or columns, recording that the determinant changes sign.
• Multiply a row or column by $k$, recording that the determinant is multiplied by $k$.
• Add a multiple of one row or column to another, which leaves the determinant unchanged.
• For a triangular or diagonal matrix, multiply the diagonal entries.

These properties permit complicated calculations to be simplified through suitable row or column operations.

5. Finding the inverse using the adjoint

For a nonsingular matrix $A$:

• Calculate $\det (A)$.
• Form every minor $M_{ij}$.
• Convert the minors into cofactors using

$$C_{ij}=(-1{)}^{i+j}{M}_{ij}.$$

• Form the cofactor matrix.
• Transpose the cofactor matrix to obtain $\mathrm{adj}(A)$.
• Divide by the determinant:

$$A^{-1}=\frac{\mathrm{adj}(A)}{|A|}.$$

• The method is valid only when $|A|\ne 0$.

For a $2\times 2$ matrix, this produces $$A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right],$$ provided $ad-bc\ne 0$.

6. Solving $AX=B$

If $A$ is invertible:

• Confirm that $\det (A)\ne 0$.
• Find $A^{-1}$.
• Multiply the equation $AX=B$ on the left by $A^{-1}$:

$$A^{-1}AX=A^{-1}B.$$

• Use $A^{-1}A=I$ to obtain

$$X=A^{-1}B.$$

7. Solving simultaneous equations using Cramer’s rule

For a system of linear equations:

• Form the coefficient determinant $D$.
• Confirm that $D\ne 0$.
• For each variable, replace its coefficient column with the constant column.
• Evaluate the resulting determinant.
• Divide by $D$. For example, a variable $x$ is given by

$$x=\frac{D_x}{D}.$$

• Repeat for each variable.

If $D\ne 0$, the system has a unique solution. If $D=0$, Cramer’s rule does not provide a unique solution, and further comparison of determinants or equations is required. The method is particularly effective for small systems involving two or three unknowns.

8. Finding the area of a triangle

For vertices $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$:

• Substitute the coordinates into

$$\text{Area}=\frac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|.$$

• Evaluate the expression inside the absolute value.
• Multiply by $\frac{1}{2}$.
• Use the absolute value to ensure that the area is nonnegative.

Where It Goes Wrong

• Forgetting the negative sign on the middle term in the $3\times 3$ expansion:

$$a(ei-fh)-b(di-fg)+c(dh-eg).$$

• Using the wrong cofactor sign instead of the checkerboard pattern

$$+-+;-+-;+-+.$$

• Applying the inverse formula when the determinant is zero; a singular matrix has no inverse.
• Confusing the minor $M_{ij}$ with the cofactor $C_{ij}=(-1{)}^{i+j}{M}_{ij}$.
• Forgetting that interchanging two rows or columns changes the sign, whereas adding a multiple of one row or column to another does not change the determinant.
• Applying Cramer’s rule without checking that the coefficient determinant is nonzero; when it is zero, the system may have no solution or infinitely many solutions.

What Gets Asked

• Calculate a $2\times 2$ or $3\times 3$ determinant.
• Expand a determinant along a specified or convenient row or column.
• Find minors and cofactors using the checkerboard sign pattern.
• Simplify a determinant using row and column properties.
• Determine whether a matrix is singular or nonsingular.
• Find the inverse of a matrix using its adjoint and determinant.
• Verify identities such as

$$A{A}^{-1}=A^{-1}A=I,\det (AB)=\det (A)\det (B),\det (A^T)=\det (A).$$

• Solve $AX=B$ using

$$X=A^{-1}B.$$

• Solve systems of two or three simultaneous linear equations using Cramer’s rule.
• Determine whether a linear system has a unique solution, no solution, or infinitely many solutions from its coefficient determinant and further equation comparisons.
• Find the area of a triangle from three coordinate vertices using the determinant formula.