2.4 Non-Singular Matrices
Definition 2.4.1. An \(n\) by \(m\) matrix is said to be non-singular if there exists a matrix \(B\),denoted by \(A^{-1}\), such that \(AB=BA=I_n\) where \(I_n\) is the \(n\) by \(n\) identity matrix.
Non-singular matrices are also called invertible matrices. A matrix which is not non-singular is called singular.
Example 2.4.2. The matrix \(A= \begin {pmatrix} -2&5\\1&-3\\ \end {pmatrix} \) is non-singular if \(B= \begin {pmatrix} -3&-5\\-1&-2\\ \end {pmatrix} \) then \(AB= \begin {pmatrix} 1&0\\0&1\\ \end {pmatrix} \) and \(BA= \begin {pmatrix} 1&0\\0&1\\ \end {pmatrix} \).
Proof. Suppose \(B\) and \(C\) are both inverses of \(A\). Then \(AB=BA=I\) and \(AC=CA=I\).
Therefore, \(B=B\cdot I=B(AC)=(BA)C=I\cdot C=C\). Thus \(B=C\). □
The next theorem gives properties of non-singular matrices.
Theorem 2.4.4. Let \(A\) and \(B\) be non-singular \(n\) by \(n\) matrices and \(k\) be a non-zero real number. Then
- 1.
- \(A^{-1}\) is non-singular and \((A^{-1})^{-1}=A\).
- 2.
- \(kA\) is non-singular and \((kA)^{-1}=\dfrac {1}{k}\,A^{-1}\).
- 3.
- \(AB\) is non-singular and \((AB)^{-1}=A^{-1}B^{-1}\).
- 4.
- \(A^t\) is non-singular and \((A^t)^{-1}=(A^{-1})^t\).
Proof.
- 1.
- Since \(A^{-1}A=AA^{-1}=I\), \(A^{-1}\) is non-singular and \((A^{-1})^{-1}=A\).
- 2.
- We have \((kA)\Big (\dfrac {1}{k}\,A^{-1}\Big )=k\cdot \dfrac {1}{k}\,A\cdot A^{-1}=I\) and \(\Big (\dfrac {1}{k}\,A^{-1}\Big )(kA)=\dfrac {1}{k}\cdot kA^{-1}A=I\).
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