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Invertible Matrix

Invertible Matrix
Nonsingular Matrix

A square matrix which has an inverse. A matrix is nonsingular if and only if its determinant does not equal zero.

 

 

See also

Singular matrix

Key Formula

AA−1=A−1A=IA A^{-1} = A^{-1} A = I
Where:
  • AA = The original square matrix
  • A−1A^{-1} = The inverse of matrix A
  • II = The identity matrix of the same size as A

Worked Example

Problem: Determine whether the matrix A is invertible, and if so, find its inverse. A = [[2, 1], [5, 3]]
Step 1: Compute the determinant of A. For a 2×2 matrix [[a, b], [c, d]], the determinant is ad − bc.
det⁡(A)=(2)(3)−(1)(5)=6−5=1\det(A) = (2)(3) - (1)(5) = 6 - 5 = 1
Step 2: Check whether the determinant is zero. Since det(A) = 1 ≠ 0, the matrix is invertible.
Step 3: Use the 2×2 inverse formula: swap the diagonal entries, negate the off-diagonal entries, and divide everything by the determinant.
A−1=1det⁡(A)[d−b−ca]=11[3−1−52]A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} = \frac{1}{1} \begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}
Step 4: Verify by multiplying A and A⁻¹ to confirm the result is the identity matrix.
AA−1=[2153][3−1−52]=[1001]=IA A^{-1} = \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix} \begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I
Answer: The matrix A is invertible, and its inverse is A⁻¹ = [[3, −1], [−5, 2]].

Another Example

Problem: Show that the matrix B = [[4, 2], [6, 3]] is not invertible.
Step 1: Compute the determinant of B.
det⁡(B)=(4)(3)−(2)(6)=12−12=0\det(B) = (4)(3) - (2)(6) = 12 - 12 = 0
Step 2: Since det(B) = 0, the matrix B is not invertible (it is singular). Notice that the second row [6, 3] is exactly 1.5 times the first row [4, 2], so the rows are linearly dependent — a hallmark of singular matrices.
Answer: B is not invertible because its determinant equals zero. Its rows are proportional, so it has no inverse.

Frequently Asked Questions

How do you tell if a matrix is invertible?
Compute the determinant of the matrix. If the determinant is not zero, the matrix is invertible. If the determinant equals zero, the matrix is singular (not invertible). Equivalently, a matrix is invertible if and only if its rows (or columns) are linearly independent.
Can a non-square matrix be invertible?
No. Only square matrices (same number of rows and columns) can have a true two-sided inverse. A non-square matrix may have a left inverse or a right inverse in special cases, but it cannot be invertible in the standard sense.

Invertible (Nonsingular) Matrix vs. Singular Matrix

An invertible matrix has a nonzero determinant, linearly independent rows/columns, and a unique inverse. A singular matrix has a determinant of zero, at least one row or column that is a linear combination of the others, and no inverse. When solving a system Ax = b, an invertible coefficient matrix A guarantees a unique solution, while a singular coefficient matrix means the system has either no solution or infinitely many solutions.

Why It Matters

Invertible matrices are essential for solving systems of linear equations: if A is invertible, the system Ax=bAx = b has the unique solution x=A−1bx = A^{-1}b. They also appear throughout science and engineering — in computer graphics transformations, cryptography, and data analysis — wherever you need to reverse a linear operation. Understanding invertibility is a gateway concept for deeper topics like eigenvalues, matrix factorizations, and linear transformations.

Common Mistakes

Mistake: Assuming every square matrix is invertible.
Correction: A square matrix is invertible only when its determinant is nonzero. Many square matrices, such as [[1, 2], [2, 4]], have determinant zero and therefore have no inverse.
Mistake: Confusing the inverse of a matrix with its transpose.
Correction: The inverse A⁻¹ satisfies A A⁻¹ = I, while the transpose Aᵀ simply swaps rows and columns. These are different operations. A matrix where the inverse equals the transpose is a special case called an orthogonal matrix.

Related Terms