Determinants

The One Thing

A determinant is a scalar associated with a square matrix that provides a compact test for invertibility and supports the evaluation of matrices, the solution of simultaneous linear equations, and geometric calculations such as triangle areas. Minors, cofactors, and determinant properties provide systematic methods for calculating determinants and constructing matrix inverses.

Definitions and Results

• Determinant: A scalar value associated with a square matrix, written using vertical bars, such as $|A|$ or $\det (A)$.
• Second-order determinant: For $\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|$,

$$|A|=ad-bc.$$

• Third-order determinant: A determinant formed from a $3\times 3$ square matrix and evaluated by cofactor expansion or determinant properties. For

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

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

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

• Cofactor sign pattern: For a $3\times 3$ determinant, the signs are

$$\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}.$$

• Expansion by cofactors: A determinant may be expanded along any row or column:

$$|A|=a_{i1}{C}_{i1}+a_{i2}{C}_{i2}+\cdots +a_{in}{C}_{in}.$$

• Singular matrix: A square matrix with determinant zero. It has no inverse.
• Non-singular matrix: A square matrix with non-zero determinant. It has an inverse.
• Adjoint or adjugate: The transpose of the cofactor matrix of a square matrix.
• Inverse of a matrix: For a non-singular matrix $A$,

$$A^{-1}=\frac{\mathrm{adj}(A)}{|A|}.$$ The inverse exists if and only if $|A|\ne 0$.

• Adjoint identity: For a square matrix $A$,

$$A\mathrm{adj}(A)=\mathrm{adj}(A)A=|A|I.$$

• Cramer’s rule: A method for solving $n$ linear equations in $n$ variables using determinants, provided the coefficient determinant is non-zero. For a three-variable system,

$$x=\frac{D_x}{D},y=\frac{D_y}{D},z=\frac{D_z}{D},$$ where $D$ is the coefficient determinant and $D_x,D_y,D_z$ are obtained by replacing the corresponding coefficient columns with the constants column.

• Homogeneous system: A system of linear equations in which all constant terms are zero. It always has the trivial solution and may have non-trivial solutions when the coefficient determinant is zero.
• Consistent system: A system of equations with at least one solution.
• Transpose property:

$$|A^T|=|A|.$$

• Row and column operations:

- 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 one row or column is zero, the determinant is zero. - Multiplying one row or one 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.

• Triangular or diagonal determinant: The determinant equals the product of the principal diagonal elements.
• Matrix equation: For $AX=B$, if $|A|\ne 0$, the unique solution is

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

• Area of a triangle: For the equations

$$a_{1}x+b_{1}y+c_1=0,a_{2}x+b_{2}y+c_2=0,a_{3}x+b_{3}y+c_3=0,$$ the area of the triangle formed by the corresponding points is one-half the absolute value of the associated $3\times 3$ determinant.

• Collinearity: Three points are collinear if the determinant used for their area is zero.
• Solvability of linear systems: A non-zero coefficient determinant guarantees a unique solution. A zero coefficient determinant requires further examination to determine consistency and the number of solutions.

Worked Methods

1. Evaluating a second-order determinant

For $$\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|,$$ multiply the principal diagonal elements and subtract the product of the other diagonal elements: $$|A|=ad-bc.$$

2. Evaluating a third-order determinant by first-row expansion

For $$\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|,$$ use the cofactor sign pattern $+,-,+$. The result is $$a(ei-fh)-b(di-fg)+c(dh-eg).$$

The associated minors are obtained by deleting the relevant row and column. The corresponding cofactors are then formed using $$C_{ij}=(-1{)}^{i+j}{M}_{ij}.$$

3. Expanding a determinant by cofactors

To expand along a chosen row or column:

• Select the row or column.
• For each element, delete its row and column to obtain its minor.
• Apply the appropriate cofactor sign.
• Multiply each original element by its cofactor.
• Add the resulting terms.

For expansion along row $i$, $$|A|=a_{i1}{C}_{i1}+a_{i2}{C}_{i2}+\cdots +a_{in}{C}_{in}.$$

4. Simplifying a determinant using row and column properties

Apply operations that preserve or predictably alter the determinant:

• Interchange rows or columns when useful, remembering that the sign changes.
• Add a multiple of one row or column to another; the determinant remains unchanged.
• Factor a scalar $k$ from a row or column; the determinant is multiplied by $k$.
• Identify identical, proportional, or all-zero rows or columns, which immediately give determinant zero.
• Reduce the matrix to triangular or diagonal form where possible.
• Multiply the principal diagonal elements to obtain the determinant.

For a triangular or diagonal matrix, the determinant is the product of the principal diagonal elements.

5. Finding the inverse using the adjoint

For a square matrix $A$:

• Calculate each 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 $\mathrm{adj}(A)$.
• Calculate $|A|$.
• Check that $|A|\ne 0$.
• Use

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

The identity $$A\mathrm{adj}(A)=\mathrm{adj}(A)A=|A|I$$ underlies this method.

6. Solving $AX=B$

For a system written as $$AX=B:$$

• Form the coefficient matrix $A$.
• Calculate $|A|$.
• If $|A|\ne 0$, calculate $A^{-1}$.
• Obtain the unique solution from

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

If $|A|=0$, the system requires further examination for consistency and the number of solutions.

7. Solving a three-variable system using Cramer’s rule

For a three-variable system:

• Form the coefficient determinant $D$.
• Replace the $x$-coefficient column with the constants column to form $D_x$.
• Replace the $y$-coefficient column with the constants column to form $D_y$.
• Replace the $z$-coefficient column with the constants column to form $D_z$.
• Provided $D\ne 0$, calculate

$$x=\frac{D_x}{D},y=\frac{D_y}{D},z=\frac{D_z}{D}.$$

8. Finding the area of a triangle and testing collinearity

For the equations $$a_{1}x+b_{1}y+c_1=0,a_{2}x+b_{2}y+c_2=0,a_{3}x+b_{3}y+c_3=0,$$ form the associated $3\times 3$ determinant.

• Evaluate the determinant.
• Take its absolute value.
• Multiply by one-half to obtain the triangle’s area.
• If the determinant is zero, the area is zero and the three points are collinear.

Where It Goes Wrong

• Confusing a minor $M_{ij}$ with a cofactor $C_{ij}$; the cofactor requires the sign factor $(-1{)}^{i+j}$.
• Forgetting the negative sign in the middle term when expanding a $3\times 3$ determinant along the first row:

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

• Treating a row swap as determinant-preserving; interchanging two rows or columns changes the sign.
• Forgetting that multiplying one row or column by $k$ multiplies the determinant by $k$, whereas adding a multiple of one row or column to another leaves it unchanged.
• Applying the inverse formula when $|A|=0$; a zero determinant means the matrix is singular and has no inverse.
• Using Cramer’s rule without checking that the coefficient determinant $D$ is non-zero.
• Omitting the absolute value or the factor one-half when calculating the area of a triangle from its determinant.

What Gets Asked

• Evaluate a second-order determinant using $ad-bc$.
• Expand a third-order determinant, particularly

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

• Find specified minors and cofactors using the $3\times 3$ cofactor sign pattern.
• Evaluate a determinant after row or column operations.
• Determine whether a matrix is singular or non-singular.
• Use determinant properties for triangular, diagonal, identical, proportional, or zero rows and columns.
• Construct the adjoint and find $A^{-1}$.
• Solve $AX=B$ using $X=A^{-1}B$.
• Solve a three-variable system using Cramer’s rule:

$$x=\frac{D_x}{D},y=\frac{D_y}{D},z=\frac{D_z}{D}.$$

• Determine whether a linear system has a unique solution, requires further consistency analysis, or may have non-trivial solutions in the homogeneous case.
• Calculate the area of a triangle using the associated $3\times 3$ determinant.
• Test whether three points are collinear by showing that the relevant determinant is zero.