Chapter 4
Algebra
24 questions

Determinants

Determinants up to order three, minors and cofactors, adjoint and inverse, area of a triangle, and solving linear systems.

Every square matrix has a determinant: a single number that answers one crucial question — can this matrix be inverted? If the determinant is zero it cannot, and any system of equations built on it either has no solution or infinitely many.

That single test is what makes the chapter useful. It settles whether three points are collinear, whether a triangle has zero area, and whether a set of simultaneous equations is consistent, all with the same computation.

What you should be able to do

Evaluate determinants of order two and three.
Compute minors and cofactors and use them to build the adjoint.
Find the inverse of a matrix using the adjoint, and state when it fails to exist.
Find the area of a triangle from the coordinates of its vertices, and test three points for collinearity.
Decide whether a system of linear equations is consistent, and solve it by the matrix method.

Topics

5 topics

Determinant of a Square Matrix
Evaluating determinants of order up to three, by expansion along a row or column.
Minors and Cofactors
Definition and computation of minors and cofactors, and expansion using them.
Area of a Triangle
Using a determinant to find the area of a triangle and to test for collinear points.
Adjoint and Inverse of a Matrix
Constructing the adjoint, and finding the inverse as the adjoint divided by the determinant.
Consistency and Solution of Linear Equations
Testing a system for consistency and solving two- and three-variable systems by the matrix method.

Formulas

13 items · All formulas

Determinant of order 2
\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc
Determinant of order 3 (expansion along the first row)
|A| = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}
Any row or column may be used. Choose the one with the most zeros — it removes whole terms.
Cofactor
C_{ij} = (-1)^{\,i+j}\, M_{ij}
M_{ij} is the minor: the determinant remaining after deleting row i and column j.
Adjoint
\operatorname{adj}(A) = \left[C_{ij}\right]^{T}
The transpose of the cofactor matrix. Forgetting the transpose is the classic error.
Inverse of a matrix
A^{-1} = \dfrac{1}{|A|}\operatorname{adj}(A)
Valid when
Exists if and only if |A| \neq 0.
Fundamental adjoint identity
A\,(\operatorname{adj} A) = (\operatorname{adj} A)\,A = |A|\,I
A quick way to check an adjoint before using it.
Determinant of the adjoint
|\operatorname{adj} A| = |A|^{\,n-1}
Valid when
A of order n. For n = 3: |\operatorname{adj} A| = |A|^{2}.
Determinant of a product
|AB| = |A|\,|B|
Determinants do commute, even though the matrices do not.
Determinant of a scalar multiple
|kA| = k^{\,n}\,|A|
Valid when
A of order n. The scalar is taken from every one of the n rows.
Determinant of an inverse
\left|A^{-1}\right| = \dfrac{1}{|A|}
Valid when
|A| \neq 0.
Area of a triangle
\Delta = \dfrac{1}{2}\left|\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}\right|
The outer absolute value matters: area is never negative.
Condition for collinear points
\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = 0
Three points are collinear exactly when the triangle they form has zero area.
Matrix method for linear systems
AX = B \;\Rightarrow\; X = A^{-1}B
Valid when
Valid when |A| \neq 0; then the solution is unique.

Key points

7 items · All key points

A^{-1} exists if and only if |A| \neq 0.
So computing |A| first tells you whether there is any point continuing.
The adjoint is the transpose of the cofactor matrix, not the cofactor matrix itself.
Expand along whichever row or column has the most zeros — each zero kills an entire cofactor term.
|kA| = k^{n}|A| for an n \times n matrix, not k|A|.
The scalar comes out of each of the n rows separately. For a 3\times3, |2A| = 8|A|.
Area must be positive — take the absolute value of the determinant before halving.
If |A| = 0, the system AX = B has either no solution or infinitely many — never exactly one.
Compute (\operatorname{adj} A)B: if it is not the zero matrix the system is inconsistent; if it is, there are infinitely many solutions.
Determinants multiply: |AB| = |A||B|, even though AB \neq BA in general.

Common mistakes

6 items · All common mistakes

Mistake
Reporting the cofactor matrix as the adjoint.
Instead
Transpose it. \operatorname{adj}(A) = [C_{ij}]^{T}.
Why
The transpose is a separate final step that adds nothing visible for a symmetric-looking matrix, so it gets dropped.
Mistake
Omitting the sign factor (-1)^{i+j} when forming cofactors.
Instead
Write out the sign grid \begin{bmatrix} + & - & + \\ - & + & - \\ + & - & +\end{bmatrix} before starting.
Why
Minors and cofactors differ only by the sign, so the two are easily conflated.
Mistake
Writing |kA| = k|A|.
Instead
|kA| = k^{n}|A| where n is the order of A.
Why
Scalar multiplication of a matrix scales every entry, and a determinant picks up one factor from each row.
Mistake
Giving a negative area for a triangle.
Instead
The determinant may come out negative; area is \frac{1}{2} of its absolute value.
Mistake
Concluding that a system with |A| = 0 has no solution.
Instead
It has no unique solution. Test (\operatorname{adj} A)B to decide between no solution and infinitely many.
Why
'No unique solution' is often read as 'no solution'.
Mistake
Computing the inverse before checking whether the determinant is zero.
Instead
Evaluate |A| first. If it is zero, state that A^{-1} does not exist and stop.

Back to all chapters

Determinants

What you should be able to do

Topics

Formulas

Determinant of order 2

Determinant of order 3 (expansion along the first row)

Cofactor

Adjoint

Inverse of a matrix

Fundamental adjoint identity

Determinant of the adjoint

Determinant of a product

Determinant of a scalar multiple

Determinant of an inverse

Area of a triangle

Condition for collinear points

Matrix method for linear systems

Key points

Common mistakes