The Inverse and Determinant of a Matrix
Introduced| English |
|---|
| inverse matrix/ɪnˈvɜːs ˈmeɪtrɪks/ |
| determinant/dɪˈtɜːmɪnənt/ |
| identity matrix/aɪˈdentɪti ˈmeɪtrɪks/ |
| not invertible/nɒt ɪnˈvɜːtɪbl/ |
Undoing a matrix
- Multiplying by a matrix transforms space — but can you undo it?
- For ordinary numbers, dividing by $5$ undoes multiplying by $5$.
- Matrices have their own "reverse" — the inverse matrix.
- First we need one number that decides whether a reverse even exists: the determinant.
The determinant
- The determinant 行列式 of $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ is the single number $ad - bc$.
- Geometrically, its size is the area of the parallelogram the columns span.
- A large determinant means the matrix stretches area a lot; a small one, very little.
- A determinant of zero means the columns line up — the parallelogram is flat.
For $\begin{bmatrix}3 & 1\\ 1 & 2\end{bmatrix}$, the determinant is $3\cdot2 - 1\cdot1$. What is it?
The determinant of $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ is $ad - bc = 3\cdot2 - 1\cdot1 = 5$.
Geometrically, the size of a 2 × 2 determinant tells you…
The absolute value of the determinant is the area scaled by the matrix — zero area means the columns are parallel.
The inverse
- The inverse matrix 逆矩阵 $A^{-1}$ is the matrix that undoes $A$.
- Multiplying them gives the identity matrix 单位矩阵 $I$ — the matrix version of $1$: $A A^{-1} = I$.
- For a $2\times2$ matrix, $A^{-1} = \dfrac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$.
- Notice the determinant $ad - bc$ sitting in the denominator.
Multiplying a matrix by its inverse gives the identity matrix.
By definition $A A^{-1} = I$, the identity matrix — the matrix version of the number $1$.
When an inverse exists
- You cannot divide by $ad - bc$ if it is zero — so a zero determinant means no inverse.
- Such a matrix is not invertible 不可逆的: it squashes the plane onto a line and loses information.
- A non-zero determinant guarantees a unique inverse.
- So the determinant is the quick test: non-zero $\Rightarrow$ invertible.
The determinant is an area factor
The determinant tells you how much a matrix scales area; here it scales area by four.
A 2 × 2 matrix has an inverse exactly when its determinant is…
If $\det = 0$ the columns are parallel and no inverse exists — the matrix is not invertible.
Select all true statements.
Only matrices with a non-zero determinant are invertible, so not every matrix has an inverse.
Why it matters
- An inverse solves a matrix equation $A\vec{x} = \vec{b}$ in one step: $\vec{x} = A^{-1}\vec{b}$.
- That is exactly how a system of linear equations is solved with matrices.
- The determinant also tells you when a system has no unique solution (det $= 0$).
- These ideas power computer graphics, cryptography, and data science.
A matrix has an inverse only when its determinant is non-zero. If $ad - bc = 0$, the formula divides by zero — there is no inverse, and a system $A\vec{x} = \vec{b}$ has either no solution or infinitely many. Always check the determinant first.
Invert $A = \begin{bmatrix}3 & 1\\ 1 & 2\end{bmatrix}$.
- Determinant: $3\cdot2 - 1\cdot1 = 5$ (non-zero, so an inverse exists).
- Swap the diagonal, negate the off-diagonal, divide by $5$:
$$A^{-1} = \frac{1}{5}\begin{bmatrix}2 & -1\\ -1 & 3\end{bmatrix}.$$
- Check: $A A^{-1} = I$. ✓
The determinant $ad - bc$ is one number measuring the area a matrix scales. The inverse $A^{-1}$ undoes the matrix, with $A A^{-1} = I$ (the identity matrix). An inverse exists only when the determinant is non-zero; a zero determinant means the matrix is not invertible.