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Determinant &
Invertibility

Discover the single number that reveals whether a matrix can be reversed: the determinant. Learn how to compute it and why it decides invertibility.

Linear Algebra

What Is a Determinant?

The determinant is a special number computed from a square matrix. For a \(2 \times 2\) matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\), the determinant is:

It tells you how the matrix scales area, whether it flips orientation, and most importantly: whether the matrix is invertible.

The Geometric Meaning

Unit Square Area = 1 A Parallelogram
Area = \(|\det(A)|\)
Before After transformation

When \(\det(A) = 0\)

Unit Square A Collapsed! Area = 0 Entire plane squashed to a line

What Is Invertibility?

A matrix \(A\) is invertible (also called nonsingular) if there exists a matrix \(A^{-1}\) such that:

Invertible
\(\det(A) \neq 0\): the matrix has an inverse, solves systems uniquely
Singular
\(\det(A) = 0\): no inverse exists, system has zero or infinitely many solutions

The Invertible Matrix Theorem

1
\(A\) is invertible
2
\(\det(A) \neq 0\)
3
The columns of \(A\) are linearly independent
4
\(A\mathbf{x} = \mathbf{b}\) has a unique solution for every \(\mathbf{b}\)
5
The reduced row echelon form of \(A\) is the identity \(I\)

The \(2 \times 2\) Determinant Formula

a b c d ad - bc
\(\det = ad - bc\)

The \(2 \times 2\) Inverse Formula

When \(\det(A) \neq 0\), the inverse of a \(2 \times 2\) matrix is:

1
Swap the main diagonal entries (\(a \leftrightarrow d\))
2
Negate the off-diagonal entries (\(b \to -b, c \to -c\))
3
Divide every entry by the determinant \(ad - bc\)

Invertible vs Singular

Property Invertible Singular
Determinant \(\neq 0\) \(= 0\)
Inverse Exists Does not exist
\(A\mathbf{x} = \mathbf{b}\) Unique solution No or infinite solutions
Column vectors Linearly independent Linearly dependent
Geometric effect Preserves dimension Collapses dimension

How to Check Invertibility

1
Identify the matrix entries. For \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\), note each value.
2
Compute \(\det(A) = ad - bc\).
3
Decide: If \(\det(A) \neq 0\), the matrix is invertible. If \(\det(A) = 0\), it is singular.
4
If invertible, find the inverse using \(A^{-1} = \frac{1}{\det(A)}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\).

Example 1: Invertible Matrix

Find the determinant of \(A = \begin{bmatrix} 3 & 1 \\ 2 & 5 \end{bmatrix}\) and decide if it is invertible.

Example 2: Singular Matrix

Find the determinant of \(B = \begin{bmatrix} 4 & 6 \\ 2 & 3 \end{bmatrix}\).

Example 3: \(3 \times 3\) Determinant

Compute \(\det\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix}\) using cofactor expansion along the first column.

Key Properties of Determinants

Product Rule
\(\det(AB) = \det(A) \cdot \det(B)\)
Transpose
\(\det(A^T) = \det(A)\)
Inverse
\(\det(A^{-1}) = \frac{1}{\det(A)}\)
Row Swap
Swapping two rows negates the determinant
Scalar Multiple
\(\det(kA) = k^n \det(A)\) for an \(n \times n\) matrix

Common Mistakes

✕ Wrong Diagonal
Computing \(ab - cd\) instead of \(ad - bc\). Remember: main diagonal first, then subtract the anti-diagonal product.
✕ Forgetting the Sign
Writing \(ad + bc\) instead of \(ad - bc\). The determinant always subtracts the anti-diagonal product.
✕ Scalar Scaling
Thinking \(\det(2A) = 2\det(A)\). For an \(n \times n\) matrix, \(\det(kA) = k^n \det(A)\), not \(k \cdot \det(A)\).

Summary Cheat Sheet

Formula / Rule Expression
\(2 \times 2\) determinant \(ad - bc\)
Invertibility test \(\det(A) \neq 0\)
\(2 \times 2\) inverse \(\frac{1}{\det(A)}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\)
Triangular matrix det Product of diagonal entries
Product rule \(\det(AB) = \det(A)\det(B)\)
Scalar rule \(\det(kA) = k^n\det(A)\)

Question 1 of 5

What is the determinant of \(\begin{bmatrix} 5 & 3 \\ 2 & 4 \end{bmatrix}\)?

Question 2 of 5

Which matrix is not invertible?

Question 3 of 5

If \(\det(A) = 5\), what is \(\det(A^{-1})\)?

Question 4 of 5

If \(A\) is a \(3 \times 3\) matrix with \(\det(A) = 4\), what is \(\det(2A)\)?

Question 5 of 5

What is the inverse of \(A = \begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix}\)?