# Matrix Inverses

Mathematics I · Matrices and Vectors · https://tryals.app/learn/mathematics-i/matrix-inverses

## Undoing a Matrix

The **inverse** of a square matrix $A$ is the matrix $A^{-1}$ that undoes it: $AA^{-1} = A^{-1}A = I$, where $I$ is the identity. 

A matrix is **invertible** (or nonsingular) if and only if **$\det A \neq 0$**. Singular matrices ($\det A = 0$) collapse space, destroying information that cannot be restored.

For a $2 \times 2$ matrix, use this formula:
$$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \Rightarrow A^{-1} = \frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$

Swap diagonal entries, negate off-diagonal ones, and divide by the determinant. Division by $\det A$ shows why zero determinants have no inverse.

| Property | Rule | Meaning |
| --- | --- | --- |
| **Uniqueness** | One $A^{-1}$ | Exists uniquely when $\det A \neq 0$ |
| **Double Inverse** | $(A^{-1})^{-1} = A$ | Undoing the undoing returns $A$ |
| **Product Rule** | $(AB)^{-1} = B^{-1}A^{-1}$ | Order reverses like shoes and socks |

## Solving and Pitfalls

The primary use of the inverse is **solving linear systems**. If $A\mathbf{x} = \mathbf{b}$ and $A$ is invertible, multiply by $A^{-1}$ to get:
$$\mathbf{x} = A^{-1}\mathbf{b}$$

This gives the unique solution directly, though Gaussian elimination is faster for large systems.

**Common Pitfalls to Avoid:**
- **Not every matrix is invertible.** Only matrices with a **nonzero determinant** have an inverse.
- **Do not split products incorrectly.** The inverse of a product is **$(AB)^{-1} = B^{-1}A^{-1}$**, not $A^{-1}B^{-1}$. 

Multiplication is non-commutative, so the undoing steps must happen in the exact reverse sequence.

## Practice questions

6 of this lesson's 10 practice questions, with answers. The full set is in the app.

### 1. Consider the matrix with rows (1, 2) and (2, 4). Select every TRUE statement about it.

A. Ax = b has a unique solution for every b
B. It is singular
C. Its rows are proportional
D. It has an inverse
E. Its determinant is 0

**Answer:** B. It is singular; C. Its rows are proportional; E. Its determinant is 0

**Why:** The determinant is 1(4) - 2(2) = 0, so the matrix is singular with no inverse, and its proportional rows are why. With a zero determinant the system Ax = b has either no solution or infinitely many, depending on b, never a unique one.

Page: https://tryals.app/practice/mathematics-i/matrix-inverses/consider-the-matrix-with-rows-1-2-and-2-4-select-every-true

### 2. Match each term to its meaning.

**Answer:**

- The identity matrix I → Leaves every vector exactly where it was
- The inverse of A → The matrix that undoes what A did
- A singular matrix → Collapses space, so nothing can undo it
- Undoing a composition → Reverse the last step first, then the one before

**Why:** The inverse undoes A, singular matrices cannot be undone, products invert in reverse order, and the inverse packages the solution of a linear system. In practice Gaussian elimination is faster for large systems, but the inverse is the cleanest way to see WHY the solution is unique.

Page: https://tryals.app/practice/mathematics-i/matrix-inverses/match-each-term-to-its-meaning

### 3. The inverse of a product reverses the order: (AB)⁻¹ = B⁻¹A⁻¹, not A⁻¹B⁻¹. Why does undoing a composition require going backwards through it?

A. Because inverses are computed by transposing, and transposition reverses order.
B. Because matrix inversion reverses the order of all operations as a formal rule, without geometric meaning.
C. Because AB means do B then A, so undoing reverses the order.
D. Because A⁻¹B⁻¹ is undefined whenever AB is defined, so only one ordering is available.

**Answer:** C. Because AB means do B then A, so undoing reverses the order.

**Why:** AB means apply B then A, so undoing it means undoing the most recent step first, socks then shoes going on, shoes then socks coming off. The algebra shows the mechanism: (AB)(B⁻¹A⁻¹) cancels from the inside out. A⁻¹B⁻¹ is perfectly well defined and simply wrong, corresponding to undoing B first.

Page: https://tryals.app/practice/mathematics-i/matrix-inverses/the-inverse-of-a-product-reverses-the-order-ab-b-a-not

### 4. Sort each statement about inverses by whether it is true or false.

**Answer:**

- True: The inverse of AB is the inverse of B times the inverse of A, The inverse of the inverse of A is A, When an inverse exists it is unique
- False: The inverse of AB is the inverse of A times the inverse of B, Every square matrix has an inverse

**Why:** Products invert in reverse order — socks then shoes, shoes off then socks off — because multiplication is not commutative. Singular matrices have no inverse at all, but where an inverse exists it is unique, and inverting twice returns the original.

Page: https://tryals.app/practice/mathematics-i/matrix-inverses/sort-each-statement-about-inverses-by-whether-it-is-true-or-false

### 5. Order the steps of solving the system Ax = b using an inverse.

**Answer:**

1. Check that the determinant of A is nonzero
2. Compute the inverse of A using the 2 by 2 formula
3. Multiply both sides on the left by the inverse of A
4. Simplify the left side to x, since the inverse of A times A is the identity
5. Read off the solution as x = the inverse of A times b

**Why:** Check invertibility, build the inverse, multiply on the left, simplify, read off x. Multiplying on the right would give A x A-inverse, which does not simplify, the side you multiply on is not a free choice for matrices.

Page: https://tryals.app/practice/mathematics-i/matrix-inverses/order-the-steps-of-solving-the-system-ax-b-using-an-inverse

### 6. There is a formula for the inverse of a 2×2 matrix and for 3×3, and nobody uses the general formula for a 10×10. Why does the explicit formula stop being useful so quickly?

A. Because the formula requires the determinant, which cannot be computed for large n
B. Because the formula is only valid up to 3×3 and fails for larger matrices
C. Because larger matrices are rarely invertible in practice
D. Because the cofactor expansion has n! terms, over three million at 10×10

**Answer:** D. Because the cofactor expansion has n! terms, over three million at 10×10

**Why:** n! against n³ is the whole story: 3.6 million terms versus a few hundred operations, and by n = 20 the formula is beyond any computer while elimination is instant. The formula is a proof that the inverse exists and a description of it, it was never an instruction for computing one.

Page: https://tryals.app/practice/mathematics-i/matrix-inverses/there-is-a-formula-for-the-inverse-of-a-2-2-matrix-and-for-3-3-and