How Matrices Work

As engineers, we are all familiar with how to multiply two matrices and the dimensional requirements for doing so. For example,

While this computation is correct, it offers limited insight. A more informative approach is to view the first matrix as a collection of column vectors:

In this interpretation, the result is expressed as a linear combination of the columns of the matrix. This viewpoint is not only more insightful, but also scales naturally to larger matrices.

Let’s take a look at a different situation:

We can again rewrite this product in a more revealing way:

This time, the matrix is interpreted as a collection of row vectors, and the result is a linear combination of those rows.

From these examples, we observe a fundamental rule of matrix multiplication:

• When a vector multiplies a matrix from the right, it acts on the columns of the matrix.

• When a vector multiplies a matrix from the left, it acts on the rows of the matrix.

Things become more interesting when we multiply two matrices. Since a matrix can be viewed either as a set of rows or as a set of columns, multiple—but equivalent—interpretations are possible.

Suppose

We want to calculate AB.

• First perspective: Columns of B

Just as when multiplying a single column vector by a matrix from the right, here B consists of two column vectors. Consequently, the product AB will also have two columns, one corresponding to each column of B.

• Second perspective: rows of A

Similarly, when multiplying a row vector by a matrix from the left, the result is a linear combination of the rows of the matrix. Since A contains two row vectors, the product AB will have two rows, one corresponding to each row of A.

• Third perspective: columns of A and rows of B

This time, we treat the left matrix (A) as a 1x3 row vector, and the right matrix (B) as a 3x1 column vector. With this interpretation, the matrix product reduces to a vector multiplication:

Naturally, performing the multiplication using the standard element-by-element definition (a₁₁ = dot product of the first row of A with the first column of B, and so on) leads to the same result.

This is significant because it reshapes our understanding of what matrices represent. A matrix can be interpreted consistently as a collection of rows, a collection of columns, or even as a structured collection of smaller block matrices. Consider the matrices

The multiplication of C and D could be performed as below

All of these perspectives—rows, columns, and blocks—are simultaneously correct. Together, they show that matrices are not merely arrays of numbers, but structured objects that represent linear transformations, linear combinations, and compositions of simpler components. The flexibility of these interpretations is not only fundamental to what matrices truly are, but also highly practical when working with real problems.

Building on this idea, let us consider a situation in which an unknown matrix A is multiplied by a known matrix B, resulting in a matrix C:

What can we infer about the matrix A? 

The conventional approach would be to compute B⁻¹ and multiply it from the right on both sides of the equation. However, using the perspective developed earlier, we can proceed more directly.

Since each row of C is a linear combination of the rows of B, with coefficients given by the corresponding row of A, we observe that the first row of C is identical to the first row of B. This implies that the first row of A must be [1  0  0]:

The same reasoning applies to the third row: since the third rows of B and C are identical, the third row of A must be [0  0  1].

For the second row, we look for a linear combination of the rows of B that produces the second row of C. One such combination is

Therefore, the matrix A is:

Notably, all of this information was extracted directly from the matrix equation, without explicitly computing any inverse. This illustrates the practical power of interpreting matrix multiplication through linear combinations.

Consider another example. What matrix, when multiplied from the right, swaps the columns of a 2×2matrix? That is,

Using the same reasoning, each column of the result must be a linear combination of the columns of the original matrix:

In engineering problems, we often work with matrices in which many entries are zero or follow clear structural patterns. Viewing matrices through these interpretive lenses—rows, columns, and blocks—provides not only computational efficiency, but also deeper insight into the structure and behavior of the systems they represent.