Invertible Matrices

An invertible matrix is a special type of square matrix that has an inverse. This concept is important because it connects matrix multiplication with the identity matrix and helps in solving systems of linear equations. Invertible matrices are also used in higher‑level topics such as linear transformations and determinants.

A square matrix A of order m × m is said to be invertible (or non-singular) if there exists another square matrix B of the same order such that

where I is the identity matrix of order m. Then B is called the inverse matrix of A, and it is denoted by \[A^{-1}\].

• Only square matrices can be invertible.

• The inverse matrix, if it exists, is unique for a given matrix.

• If A is invertible, then \[A^{-1}\] satisfies both \[AA^{-1} = I\] and \[A^{-1}A = I\].

• Rectangular matrices do not possess inverses.

The inverse of a square matrix, if it exists, is unique.

Proof: Let A = [aᵢⱼ] be a square matrix of order m. If possible, let B and C be two inverses of A. We shall show that B = C.

Since B is the inverse of A

AB = BA = I ...(1)

Since C is also the inverse of A

AC = CA = I ...(2)

Thus

B = BI = B(AC) = (BA)C = IC = C

So B = C, which means the inverse of A is unique.

If A and B are invertible matrices of the same order, then
(AB)⁻¹ = B⁻¹A⁻¹.

Proof: From the definition of the inverse of a matrix, we have

(AB)(AB)⁻¹ = I

or A⁻¹(AB)(AB)⁻¹ = A⁻¹ (Pre multiplying both sides by A⁻¹)

or (A⁻¹A)B(AB)⁻¹ = A⁻¹ (Since A⁻¹A = I)

or IB(AB)⁻¹ = A⁻¹

or B(AB)⁻¹ = A⁻¹

or B⁻¹B(AB)⁻¹ = B⁻¹A⁻¹

or I(AB)⁻¹ = B⁻¹A⁻¹

Hence (AB)⁻¹ = B⁻¹A⁻¹

let \[\mathbf{A}= \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}\] and \[\mathbf{B}= \begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix}\] be two matrices.

Now \[\mathrm{A}\mathrm{B}= \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix}\]

\[= \begin{bmatrix} 4-3 & -6+6 \\ 2-2 & -3+4 \end{bmatrix}= \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}=\mathrm{I}\]

Also \[\mathrm{BA}= \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}=\mathrm{I}.\] Thus, B is the inverse of A, in other words, B = A⁻¹, and A is the inverse of B, i.e., A = B⁻¹

• Invertible matrices must be square.

• The inverse satisfies \[AA^{-1} = A^{-1}A = I\].

• The inverse, if it exists, is unique.

• For invertible matrices A and B of the same order: \[(AB)^{-1} = B^{-1}A^{-1}\].

• Rectangular matrices do not have inverses.

• If B is the inverse of A, then A is also the inverse of B.