The Product of Two Nonsingular Matrices is Nonsingular

Ohio State University exam problems and solutions in mathematics

Problem 479

Prove that if $n\times n$ matrices $A$ and $B$ are nonsingular, then the product $AB$ is also a nonsingular matrix.

(The Ohio State University, Linear Algebra Final Exam Problem)
 
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Definition (Nonsingular Matrix)

An $n\times n$ matrix is called nonsingular if the only solution $\mathbf{x}\in \R^n$ of the equation $A\mathbf{x}=\mathbf{x}$ is $\mathbf{x}=\mathbf{0}$.

Proof.

We give two proofs. The first one uses a property of the determinants of matrices, and the second one uses the definition of nonsingular matrices.

Proof 1. (Using Determinant)

Recall that a matrix is nonsingular if and only if its determinant is not zero.

Since $A$ and $B$ are nonsingular, we know that
\[\det(A)\neq 0 \text{ and } \det(B) \neq 0.\] Then we have using the multiplicative property of the determinant
\begin{align*}
\det(AB)=\det(A)\det(B)\neq 0.
\end{align*}

Since the determinant of the product $AB$ is not zero, we conclude that $AB$ is a nonsingular matrix.

Proof 2. (Using Definition of Nonsingular Matrices)

Suppose that $A, B$ are nonsingular matrices.
This means that if $A\mathbf{x}=\mathbf{0}$ for some the vector $\mathbf{x}\in \R^n$, then we must have $\mathbf{x}=\mathbf{0}$.
Same for $B$.

Suppose that we have $(AB)\mathbf{x}=\mathbf{0}$ for some vector $\mathbf{x}\in \R^n$.
Let $\mathbf{v}=B\mathbf{x}\in \R^n$. Then we have
\[A\mathbf{v}=AB\mathbf{x}=\mathbf{0}.\] So since $A$ is a nonsingular matrix, we have $\mathbf{v}=\mathbf{0}$, namely, $B\mathbf{x}=\mathbf{0}$.

Since $B$ is nonsingular, this further implies that $\mathbf{x}=\mathbf{0}$.

In summary, whenever $(AB)\mathbf{x}=\mathbf{0}$, we have $\mathbf{x}=\mathbf{0}$.
Therefore, the matrix $AB$ is nonsingular.

Final Exam Problems and Solution. (Linear Algebra Math 2568 at the Ohio State University)

This problem is one of the final exam problems of Linear Algebra course at the Ohio State University (Math 2568).

The other problems can be found from the links below.

  1. Find All the Eigenvalues of 4 by 4 Matrix
  2. Find a Basis of the Eigenspace Corresponding to a Given Eigenvalue
  3. Diagonalize a 2 by 2 Matrix if Diagonalizable
  4. Find an Orthonormal Basis of the Range of a Linear Transformation
  5. The Product of Two Nonsingular Matrices is Nonsingular (This page)
  6. Determine Whether Given Subsets in ℝ4 R 4 are Subspaces or Not
  7. Find a Basis of the Vector Space of Polynomials of Degree 2 or Less Among Given Polynomials
  8. Find Values of $a , b , c$ such that the Given Matrix is Diagonalizable
  9. Idempotent Matrix and its Eigenvalues
  10. Diagonalize the 3 by 3 Matrix Whose Entries are All One
  11. Given the Characteristic Polynomial, Find the Rank of the Matrix
  12. Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$
  13. Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$

Related Question.

The converse statment is also true:

Let $A, B$ be $n\times n$ matrices and suppose $AB$ is nonsingular. Then $A$ and $B$ are nonsingular.

See the post ↴
Two Matrices are Nonsingular if and only if the Product is Nonsingular


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