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Verifying Solutions
& Matrix Identities

Learn how to check solutions to linear systems and verify fundamental matrix identities that form the backbone of linear algebra.

Linear Algebra

What Does Verify Mean?

In linear algebra, verifying means substituting a proposed answer back into the original equation and confirming both sides are equal. If you claim \(\mathbf{x}\) solves \(A\mathbf{x} = \mathbf{b}\), you multiply \(A\) by \(\mathbf{x}\) and check that the result equals \(\mathbf{b}\).

Why Verification Matters

✓ Catches Errors
Row reduction involves many arithmetic steps. A single sign error can cascade through the entire solution.
★ Builds Intuition
Substituting back helps you see how the matrix transforms the solution vector into the target vector.
★ Required on Exams
Many problems explicitly ask you to verify, and partial credit depends on showing the check.

Two Types of Verification

Checking a Solution
Given a specific vector \(\mathbf{x}\), verify that \(A\mathbf{x} = \mathbf{b}\). This tests one particular answer.
Verifying an Identity
Confirm a general relationship like \(A(BC) = (AB)C\) or \((AB)^{-1} = B^{-1}A^{-1}\). This holds for all valid matrices.

Core Rule: Checking \(A\mathbf{x} = \mathbf{b}\)

Matrix
\( A \)
 × Vector
\( \mathbf{x} \)
 = Vector
\( \mathbf{b} \)
 ? Match ✓ Compute the left side. If it equals the right side, the solution is verified.

Key Matrix Identities

1
Identity matrix: \(AI = IA = A\) for any matrix \(A\)
2
Inverse property: \(AA^{-1} = A^{-1}A = I\)
3
Transpose of product: \((AB)^T = B^T A^T\)
4
Inverse of product: \((AB)^{-1} = B^{-1}A^{-1}\)
5
Associativity: \(A(BC) = (AB)C\)

Caution: Order Matters

Scalars (numbers) Matrices
Commutativity \(ab = ba\) ✓ Always \(AB \neq BA\) ✕ Usually
Inverse of product \((ab)^{-1} = a^{-1}b^{-1}\) \((AB)^{-1} = B^{-1}A^{-1}\) (reversed!)
Transpose of product N/A \((AB)^T = B^T A^T\) (reversed!)
Zero product rule \(ab=0 \Rightarrow a=0 \text{ or } b=0\) \(AB=O\) does not imply \(A=O \text{ or } B=O\)

Step-by-Step Verification

1
Write down the equation. Identify \(A\), \(\mathbf{x}\), and \(\mathbf{b}\) (or the identity you need to verify).
2
Compute the left-hand side. Perform the matrix multiplication carefully, entry by entry.
3
Compute the right-hand side (if it is not already given as a simple vector or matrix).
4
Compare entry by entry. Every corresponding entry must be equal.
5
State your conclusion. "Since LHS = RHS, the solution is verified" or "the identity holds."

Example 1: Checking \(A\mathbf{x} = \mathbf{b}\)

Verify that \(\mathbf{x} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}\) is a solution to:

$$\begin{pmatrix} 3 & 1 \\ 2 & -4 \end{pmatrix}\begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 5 \\ 8 \end{pmatrix}$$

Example 2: A \(3 \times 3\) System

Verify \(\mathbf{x} = \begin{pmatrix} 1 \\ 0 \\ -2 \end{pmatrix}\) solves \(A\mathbf{x} = \mathbf{b}\) where:

$$A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & -1 \\ 4 & 0 & 2 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} -5 \\ 2 \\ 0 \end{pmatrix}$$

Example 3: Verifying \(AA^{-1} = I\)

Let \(A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}\). Its inverse is \(A^{-1} = \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}\). Verify \(AA^{-1} = I\).

Example 4: \((AB)^T = B^T A^T\)

Let \(A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\) and \(B = \begin{pmatrix} 0 & 5 \\ 1 & 6 \end{pmatrix}\). Verify the transpose-of-product identity.

Visualizing the Verification Flow

STEP 1
Identify \(A, \mathbf{x}, \mathbf{b}\)
 STEP 2
Compute \(A\mathbf{x}\)
 STEP 3
Compare to \(\mathbf{b}\)
 Match? Yes ✓ Verified No ✕ Not a solution

Common Mistakes

✕ Wrong Order
Writing \((AB)^{-1} = A^{-1}B^{-1}\) instead of \(B^{-1}A^{-1}\). The order must reverse.
✕ Arithmetic Slips
Dropping a negative sign in a dot product: \(2(-3) = -6\), not \(6\). Check every multiplication.
✕ Assuming Commutativity
Treating \(AB = BA\). Matrix multiplication is generally not commutative.

Summary Table

Task What You Do What You Show
Check \(A\mathbf{x}=\mathbf{b}\) Multiply \(A\) by \(\mathbf{x}\) Result equals \(\mathbf{b}\)
Verify \(AA^{-1}=I\) Multiply \(A\) by \(A^{-1}\) Result is the identity matrix \(I\)
Verify \((AB)^T = B^T A^T\) Compute both sides Both sides are equal
Verify \((AB)^{-1} = B^{-1}A^{-1}\) Compute both sides Both sides are equal

Question 1 of 5

To verify that \(\mathbf{x}\) is a solution to \(A\mathbf{x} = \mathbf{b}\), what should you do?

Question 2 of 5

Given \(A = \begin{pmatrix} 1 & 3 \\ 2 & 5 \end{pmatrix}\) and \(\mathbf{x} = \begin{pmatrix} 4 \\ -1 \end{pmatrix}\), what is \(A\mathbf{x}\)?

Question 3 of 5

What is \((AB)^{-1}\) equal to?

Question 4 of 5

If \(A = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}\), what is \(AI\) where \(I\) is the \(2 \times 2\) identity?

Question 5 of 5

If \(AB = O\) (the zero matrix), can we conclude that \(A = O\) or \(B = O\)?