Inverse Matrix Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

medium
Find the inverse of A=[2153]A = \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix}.

Solution

  1. 1
    Step 1: det⁡(A)=2(3)−1(5)=6−5=1\det(A) = 2(3) - 1(5) = 6 - 5 = 1.
  2. 2
    Step 2: Swap aa and dd, negate bb and cc: 11[3−1−52]\frac{1}{1}\begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}.
  3. 3
    Step 3: A−1=[3−1−52]A^{-1} = \begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}.
  4. 4
    Check: AA−1=[6−5−2+215−15−5+6]=[1001]AA^{-1} = \begin{bmatrix} 6-5 & -2+2 \\ 15-15 & -5+6 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} ✓

Answer

[3−1−52]\begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}
The 2×22 \times 2 inverse formula swaps the diagonal entries, negates the off-diagonal entries, and divides by the determinant. The check AA−1=IAA^{-1} = I confirms correctness.

About Inverse Matrix

The inverse of a square matrix AA, written A−1A^{-1}, is the unique matrix such that AA−1=A−1A=IAA^{-1} = A^{-1}A = I (the identity matrix). A matrix has an inverse if and only if its determinant is nonzero.

Learn more about Inverse Matrix →

More Inverse Matrix Examples

Example 2 easy

Find the inverse of [formula].

Example 3 easy

Find the inverse of [formula].

Example 4 hard

Does [formula] have an inverse? Explain.

← All Inverse Matrix Examples Inverse Matrix Concept