A Worked 2 by 2 Inverse by Row Reduction

Why Augment the Matrix?

For a two-by-two matrix, an inverse is another two-by-two matrix that undoes multiplication by the original matrix. Row reduction gives a systematic way to find that undoing matrix: start with the original matrix on the left and the identity matrix on the right, then apply every row operation to the whole array. When the left half becomes the identity, the right half records the inverse.

This method is useful because it computes the inverse through operations already used to solve linear systems. It also keeps track of how the rows of the original matrix must be combined to produce the identity. Throughout, matrices have real entries, and every example below reaches the identity on the left without requiring a row exchange.

For a two-by-two matrix, the starting layout is

The vertical bar separates the original matrix from the identity; it does not change the row-operation rules. Each operation must be applied to every entry in the row, on both sides of the bar. In particular, changing only the left half would lose the record of how the matrix was transformed.

Why the Right Half Is the Inverse

Proof. Each row operation is left multiplication by an elementary matrix: the elementary matrix is obtained by performing that operation on \(\mathbf{I}_2\). Applying a sequence of row operations therefore amounts to left multiplying both halves by the same product \(\mathbf{P}\) of elementary matrices. Thus the final augmented matrix is \([\mathbf{P}\mathbf{A}\mid\mathbf{P}]\). Since it is also \([\mathbf{I}_2\mid\mathbf{B}]\), we have \(\mathbf{P}\mathbf{A}=\mathbf{I}_2\) and \(\mathbf{P}=\mathbf{B}\). Because \(\mathbf{A}\) is invertible, \(\mathbf{A}\mathbf{A}^{-1}=\mathbf{I}_2\). Hence \(\mathbf{P}=\mathbf{P}(\mathbf{A}\mathbf{A}^{-1})=(\mathbf{P}\mathbf{A})\mathbf{A}^{-1}=\mathbf{I}_2\mathbf{A}^{-1}=\mathbf{A}^{-1}\). Therefore \(\mathbf{B}=\mathbf{A}^{-1}\). \(\square\)

The row-reduction work is therefore not a guess about the entries of the inverse. The operations construct a matrix \(\mathbf{P}\) that takes \(\mathbf{A}\) to \(\mathbf{I}_2\), while the same operations turn the right-hand identity into \(\mathbf{P}\). The theorem identifies that accumulated matrix as the inverse. The examples also verify the result directly by multiplication.

A First Calculation

Two More Row-Reduction Patterns

A Reliable Workflow and a Common Mistake

The same sequence of decisions applies to each example. The specific arithmetic changes, but the bookkeeping does not: every row operation acts on both halves, and the goal is the identity on the left.

Common mistake: row-reducing only the original matrix. For instance, start with \(\mathbf{C}=\begin{bmatrix}1&1\\0&1\end{bmatrix}\). The operation \(R_1\to R_1-R_2\) changes \(\mathbf{C}\) to \(\mathbf{I}_2\). A wrong conclusion is that the inverse is \(\mathbf{I}_2\), simply because that is what the reduced original matrix became. But \(\mathbf{C}\mathbf{I}_2=\mathbf{C}\ne\mathbf{I}_2\), so this cannot be the inverse.

The correction is to apply the same operation to the identity in the augmented matrix:

This gives the candidate \(\mathbf{C}^{-1}=\begin{bmatrix}1&-1\\0&1\end{bmatrix}\). Directly, \(\mathbf{C}\mathbf{C}^{-1}=\begin{bmatrix}1&0\\0&1\end{bmatrix}\) and \(\mathbf{C}^{-1}\mathbf{C}=\begin{bmatrix}1&0\\0&1\end{bmatrix}\). The right half matters because it records the row operations; the reduced left half alone does not record their effect on the identity.

Interpreting the Result

A computed inverse has two useful checks. First, the augmented reduction must have the exact form \([\mathbf{I}_2\mid\mathbf{B}]\); a left half that is not the identity does not yet allow the right half to be read off as the inverse. Second, multiplication should return the identity. Since the definition of an inverse requires both \(\mathbf{A}\mathbf{B}=\mathbf{I}_2\) and \(\mathbf{B}\mathbf{A}=\mathbf{I}_2\), checking both orders is a particularly direct confirmation of the candidate.

Keep the row-operation record legible. A useful practice is to name each operation immediately before or after its matrix step and to use the updated rows in subsequent calculations. Fractions are not a reason to abandon exact arithmetic: as the third example shows, a fractional pivot can be scaled to \(1\), and the final entries can still be integers. The main discipline is to preserve the same operation on both sides of the augmentation bar.