Mixed Matrix Algebra Calculation Workshop

A Reliable Order for Mixed Matrix Expressions

A matrix expression can involve several familiar operations at once: addition, scalar multiplication, multiplication, transposition, and powers. The main challenge is often not a difficult computation but keeping track of which operations are defined and which rearrangements are valid. Matrices are not numbers: in general, \(\mathbf{A}\mathbf{B}\ne\mathbf{B}\mathbf{A}\), and changing the order of factors can change the answer.

For a matrix in \(\mathbb{R}^{m\times n}\), its dimensions are \(m\times n\). Addition requires matrices of the same dimensions. The product of an \(m\times n\) matrix and a \(p\times q\) matrix is defined only when \(n=p\), and then has dimensions \(m\times q\). A transpose exchanges rows and columns, so an \(m\times n\) matrix has an \(n\times m\) transpose. A power \(\mathbf{A}^k\) is defined when \(\mathbf{A}\) is square and \(k\) is a nonnegative integer; for \(k\geq1\), it means \(k\) copies of \(\mathbf{A}\) multiplied together, while \(\mathbf{A}^0=\mathbf{I}\) of the same size.

For instance, if \(\mathbf{A}\) is \(2\times3\) and \(\mathbf{B}\) is \(3\times4\), then \(\mathbf{A}\mathbf{B}\) is defined and is \(2\times4\). But \(\mathbf{B}\mathbf{A}\) is not defined, because the inner dimensions would be \(4\) and \(2\). Even for square matrices, where both products may exist, their values need not agree. These checks should happen before any arithmetic.

Dimension Checks and Algebraic Rules

The dimension rules explain why some expressions are meaningful and others are not. They also help catch an error before it spreads through a longer calculation.

Proof. The entry in position \((i,j)\) of a sum is \(a_{ij}+b_{ij}\), so both matrices must have an entry at every same position. This requires equal numbers of rows and equal numbers of columns. For a product, its \((i,j)\) entry is \(\sum_{k=1}^{n}a_{ik}b_{kj}\). Each term requires a column index \(k\) in \(\mathbf{A}\) to match a row index \(k\) in \(\mathbf{B}\); this is possible exactly when the number of columns of \(\mathbf{A}\) equals the number of rows of \(\mathbf{B}\). The resulting entry is indexed by a row \(i\) of \(\mathbf{A}\) and a column \(j\) of \(\mathbf{B}\), giving \(m\) rows and \(q\) columns. Transposition interchanges the row and column indices, so the dimensions become \(n\times m\). \(\square\)

Several identities are especially useful in mixed expressions. Each identity below is understood only when all displayed operations are defined. Multiplication is associative, so parentheses in a product may be regrouped without changing factor order; this does not permit swapping the factors.

Proof. Write the entries of the matrices as \(a_{ij},b_{ij},c_{ij}\). For left distributivity, the \((i,j)\) entry of \((\mathbf{A}+\mathbf{B})\mathbf{C}\) is \(\sum_k(a_{ik}+b_{ik})c_{kj}\), which equals \(\sum_k a_{ik}c_{kj}+\sum_k b_{ik}c_{kj}\) by distributivity of real numbers. This is the \((i,j)\) entry of \(\mathbf{A}\mathbf{C}+\mathbf{B}\mathbf{C}\). The right-distributive identity follows because the \((i,j)\) entry of \(\mathbf{A}(\mathbf{B}+\mathbf{C})\) is \(\sum_k a_{ik}(b_{kj}+c_{kj})=\sum_k a_{ik}b_{kj}+\sum_k a_{ik}c_{kj}\). For associativity, the \((i,j)\) entry of either grouping is \(\sum_k\sum_{\ell}a_{i\ell}b_{\ell k}c_{kj}\); changing the order of these finite sums gives the same value. For the transpose identity, the \((i,j)\) entry of \((\mathbf{A}\mathbf{B})^{\top}\) is \((\mathbf{A}\mathbf{B})_{ji}=\sum_k a_{jk}b_{ki}\), which is exactly the \((i,j)\) entry of \(\mathbf{B}^{\top}\mathbf{A}^{\top}\). Finally, the power identity holds for \(k=0\) because \(\mathbf{I}^{\top}=\mathbf{I}\). For \(k\geq1\), applying the transpose-of-a-product identity repeatedly reverses the order of \(k\) identical factors, leaving \(k\) factors \(\mathbf{A}^{\top}\). \(\square\)

Worked Calculations with Dimensions Shown

Why Matrix Order Cannot Be Ignored

For real numbers, \((a+b)^2=a^2+2ab+b^2\), because \(ab=ba\). For matrices, distributivity instead gives \((\mathbf{A}+\mathbf{B})^2=\mathbf{A}^2+\mathbf{A}\mathbf{B}+\mathbf{B}\mathbf{A}+\mathbf{B}^2\). The middle terms may be combined as \(2\mathbf{A}\mathbf{B}\) only if \(\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A}\). That condition must be checked; it cannot be presumed.

Common mistake. A tempting but incorrect calculation for the matrices above is to replace \((\mathbf{A}+\mathbf{B})^2\) by \(\mathbf{A}^2+2\mathbf{A}\mathbf{B}+\mathbf{B}^2\). In this case, that proposed result is

But the direct calculation gave \((\mathbf{A}+\mathbf{B})^2=\begin{bmatrix}9&14\\0&16\end{bmatrix}\), so the proposed expansion is wrong. The correction is to retain both cross-products: \(\mathbf{A}\mathbf{B}+\mathbf{B}\mathbf{A}=\begin{bmatrix}6&6\\1&8\end{bmatrix}\). Adding this to \(\mathbf{A}^2+\mathbf{B}^2=\begin{bmatrix}3&8\\-1&8\end{bmatrix}\) gives \(\begin{bmatrix}9&14\\0&16\end{bmatrix}\).

A useful working sequence for a longer expression is:

These habits matter because a valid calculation is a chain of defined operations, not merely a sequence of plausible-looking symbols. Dimension checks detect undefined expressions; order checks detect false rearrangements; and a direct multiplication check helps identify arithmetic slips. With those safeguards, a complicated expression can be reduced to manageable sums and products without relying on numerical shortcuts that do not hold for matrices.