Training Linear Algebra Determinants and Inverses
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Determinants and Inverses

24 min Linear Algebra

Determinants and Inverses

The determinant of a square matrix is a single number that encodes an enormous amount of information. It tells you whether the matrix is invertible, how linear transformations scale area (or volume), and whether a system of equations has a unique solution. For a 2×2 matrix, the determinant is computed with a simple cross-product formula; for larger matrices the calculation is more involved, but the conceptual role remains the same.

If the determinant of a matrix is nonzero, the matrix has an inverse — another matrix that "undoes" its effect. Multiplying a matrix by its inverse produces the identity matrix, just as multiplying a number by its reciprocal gives 1. The inverse is especially useful for solving systems of equations: if $A\mathbf{x} = \mathbf{b}$, then $\mathbf{x} = A^{-1}\mathbf{b}$, and the solution can be computed in one step.

When the determinant is zero, the matrix is called singular. Singular matrices collapse space in at least one direction — they map an area to a line or a point — and no inverse exists. This corresponds to a system of equations that either has no solution or has infinitely many solutions. Recognizing singularity is therefore a critical diagnostic skill.

In this lesson you will practice computing 2×2 determinants, testing invertibility, and finding inverses using the closed-form formula. The interactive tool below lets you experiment with different matrix entries and immediately see how the determinant and inverse respond.

2x2 Determinant

For $$A=\begin{bmatrix}a&b\\c&d\end{bmatrix},$$ $$\det(A)=ad-bc.$$

Example 1

Find the determinant of $$\begin{bmatrix}2&3\\1&4\end{bmatrix}.$$

  1. $2(4)-3(1)=5$.
Invertibility Test

A square matrix is invertible if and only if its determinant is nonzero.

2x2 Inverse

If $$A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$$ and $ad-bc \ne 0$, then $$A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}.$$

Example 2

Find the inverse of $$\begin{bmatrix}1&2\\3&4\end{bmatrix}.$$

  1. Determinant $=-2$
  2. so $$A^{-1}=\frac{1}{-2}\begin{bmatrix}4&-2\\-3&1\end{bmatrix}. $$
Interactive Explorer: 2×2 Determinant & Inverse
Matrix: [2, 3; 1, 4]
det(A) = ad − bc = 5
Invertible? Yes
A⁻¹ = [0.8, −0.6; −0.2, 0.4]

Practice Problems

1. Find $$\det\begin{bmatrix}5&1\\2&3\end{bmatrix}.$$
2. Is $$\begin{bmatrix}2&4\\1&2\end{bmatrix}$$ invertible?
3. State the condition for invertibility.
4. What is the determinant of the identity matrix?
5. Why can a zero determinant be a problem for solving systems?
Show Answer Key

1. $13$

2. No; determinant is $0$

3. Determinant not equal to zero

4. $1$

5. The coefficient matrix is singular, so a unique solution does not exist.

Systems and Row Reduction Vectors and Dot Products