Checking a Proposed Matrix Inverse

What It Means to Check an Inverse

A proposed inverse is not confirmed because its entries look plausible or because a calculation used to produce them seems convincing. The check is direct: multiply the original matrix by the candidate in each order, and compare both results with the identity matrix. This is useful whether the candidate came from row reduction, a formula, or someone else’s calculation. It also makes a wrong candidate informative: the entries that fail to match the identity show exactly where the proposed relationship breaks down.

Throughout, matrices are over \(\mathbb{R}\). For a square matrix \(\mathbf{A}\) of size \(n\times n\), the identity matrix \(\mathbf{I}_n\) has \(1\)s on its main diagonal and \(0\)s elsewhere. A candidate inverse must also be an \(n\times n\) matrix. The ordinary inverse of a square matrix is defined by requiring products in both orders to equal \(\mathbf{I}_n\).

The two products involve the same matrices but are different calculations. Matrix multiplication is generally not commutative: changing the order of the factors can change the result. Therefore, the check should display \(\mathbf{A}\mathbf{B}\) and \(\mathbf{B}\mathbf{A}\) separately, rather than silently treating them as interchangeable. The required outcome in each calculation is an identity matrix of the correct size.

A Reliable Product Check

For matrices \(\mathbf{A}=(a_{ij})\) and \(\mathbf{B}=(b_{ij})\), the entry in row \(i\), column \(j\) of their product is obtained by multiplying row \(i\) of the first factor by column \(j\) of the second and adding the results. In symbols, when the product is defined,

Thus, for \(\mathbf{A}\mathbf{B}\) to equal \(\mathbf{I}_n\), each diagonal entry must be \(1\), and every off-diagonal entry must be \(0\). The same entry conditions apply to \(\mathbf{B}\mathbf{A}\), but its entries use rows of \(\mathbf{B}\) and columns of \(\mathbf{A}\). A product that is close to the identity, or has only one incorrect entry, is not the identity.

Proof. By definition, \(\mathbf{B}\) is an inverse of \(\mathbf{A}\) exactly when the two equations \(\mathbf{A}\mathbf{B}=\mathbf{I}_n\) and \(\mathbf{B}\mathbf{A}=\mathbf{I}_n\) both hold. If \(\mathbf{B}\) is an inverse, those equations therefore hold. Conversely, if both equations hold, they satisfy the definition of \(\mathbf{B}\) being an inverse of \(\mathbf{A}\). Hence the stated condition is necessary and sufficient. \(\square\)

Proof. By the definition of matrix multiplication, the \((i,j)\) entry of \(\mathbf{A}\mathbf{B}\) is \(\sum_k a_{ik}b_{kj}\). By the definition of the identity matrix, its \((i,j)\) entry is \(1\) when \(i=j\) and \(0\) when \(i\ne j\). Two matrices are equal exactly when their corresponding entries are equal, so the stated entry conditions are equivalent to \(\mathbf{A}\mathbf{B}=\mathbf{I}_n\). Applying the same reasoning to \(\mathbf{B}\mathbf{A}\) gives the second condition. \(\square\)

These results explain why a verification needs actual multiplication, not just a visual inspection of the candidate. In practice, first check that the matrices have compatible square dimensions. Then compute each product carefully, compare every entry with the identity, and conclude that the candidate is an inverse only if both equations hold.

Worked Checks with Exact Arithmetic

When a Candidate Fails

This example also illustrates how a multiplication error can be detected and corrected. The lower-right entry of \(\mathbf{A}\mathbf{C}\) is \(3\cdot(-2)+7\cdot2=-6+14=8\). Treating it as \(1\) would be an arithmetic mistake: the two products in that row-column calculation must be added exactly as written. Correcting the arithmetic shows that the candidate fails. It does not by itself establish what the correct inverse is; it establishes only that this proposed matrix is not one.

Common Pitfalls and a Careful Conclusion

Pitfall: checking only selected entries. In the last example, the upper-left entry of \(\mathbf{A}\mathbf{C}\) is \(1\), but that single match does not make the product an identity matrix. A product equals \(\mathbf{I}_n\) only if every diagonal entry is \(1\) and every off-diagonal entry is \(0\).

Pitfall: changing the order without recalculating. To find \(\mathbf{A}\mathbf{B}\), use rows of \(\mathbf{A}\) and columns of \(\mathbf{B}\). To find \(\mathbf{B}\mathbf{A}\), use rows of \(\mathbf{B}\) and columns of \(\mathbf{A}\). The first product cannot simply be copied as the second; the worked 3 by 3 example shows that the two calculations have different entry expressions even when they both end at the identity.

Pitfall: accepting a near match. An entry of \(8\) is not an entry of \(1\), and an entry of \(2\) is not an entry of \(0\). For exact matrix arithmetic, the requirement is equality, not closeness. When checking by hand, write out each row-by-column calculation so signs and arithmetic can be inspected.

A sound conclusion should name what was checked. If both \(\mathbf{A}\mathbf{B}\) and \(\mathbf{B}\mathbf{A}\) have been computed and both equal \(\mathbf{I}_n\), the definition confirms that \(\mathbf{B}\) is an inverse of \(\mathbf{A}\). If either computed product differs from \(\mathbf{I}_n\), the candidate does not meet the two-sided definition. This makes multiplication a direct test of a proposed inverse, and each entry provides a checkable record of the result.