An Introduction to Matrices

What is the difference between "matrix" and "matrices"?

The word matrices (MAY-truh-seez) is the plural of matrix; one matrix, two matrices.

What are matrices used for, in real life?

Matrices are used, mostly in the back end, in all sorts of applications of statistics, probability, search engines, optics, and the like. Most students will not encounter applied matrices in their studies, but matrices impact people every day, such as in airline scheduling and three-dimensional graphics in video games. (See StackExchange, Wikipedia, FreeCodeCamp, InfinityLearn, MIT, or a Google search, for further information.)

What is the notation for matrices?

Matrices are typically named with capital letters, such as A or C. A matrix's grid of numbers is enclosed within square brackets:

Do not use parentheses, curly braces, or vertical lines (like absolute-value notation) to enclose the grid, and certainly do not leave the grid just floating in space with no delimiters at all.

What is the size of a matrix? What are its dimensions?

The size of a matrix is the number of rows, by the number of columns. You can think of this as height-by-width. The following matrix has three rows and five columns, so its size is 3-by-5, or 3×5:

The dimensions 3×5 are read as "three by five", with the × symbol being pronounced as "by", not "times". Do not multiply the dimensions; saying that the above matrix "has size 15" is meaningless.

How are the numbers in a matrix named or referred to?

The numbers inside a matrix are called "entries", and are referenced, via subscripts, by their row and then their column. These subscripts are on letters, usually being the lower-case form of whatever upper-case letter is being used as the name of the matrix. For instance, suppose we'd named the following matrix as A:

Then the individual entries would be denoted as a_r,c, where r is the number of the row and c is the number of the column. The number 17, being in the second row and the third column of the matrix above, is the a_2,3 (pronounced as "the A-two-three" or "the A-sub-two-three") entry of the matrix A. The entry a_3,4 is the number in the third row and the fourth column; for the matrix above, a_3,4 = 6.

For smaller matrices (that is, for those with fewer than ten rows and columns), the comma in the subscript is sometimes omitted. For instance, "a_1,3 = 3" might be written as "a₁₃ = 3". This obviously won't work for larger matrices, since "a₂₁₃" would be unclear. (Does it indicate the 21,3-entry or the 2,13-entry?) It is probably a good idea, regardless of the notation used in your book, to use commas in your subscripts, for clarity's sake.

What are some types of matrices?

The entries of some types of matrices have qualities which give names to their matrices; some of these types of matrices are triangular matrices, diagonal matrices, and identity matrices.

For instance, a matrix which has the same number of rows as of columns, and whose number grid therefore forms a square, is called a "square" matrix. The top-left-to-lower-right diagonal line of numbers is called "the diagonal". The diagonal only exists for square matrices.

A matrix like the one below, with all-zero entries below the diagonal is called an "upper triangular" matrix, because all of the "interesting" entries are on the diagonal or else are above that diagonal:

(You can have lower triangular matrices, too, but I've never seen them come up, so "triangular", without the "upper" or "lower" designator, is generally taken to mean "upper triangular".)

A matrix with non-zero entries only on the diagonal is called a "diagonal" matrix.

A diagonal matrix whose diagonal entries are all 1's (no zeroes or other values) is called an "identity" matrix, for reasons which will become clear when you learn how to multiply matrices.

There are many identity matrices. The previous example was the 3×3 identity; below is the 4×4 identity:

The 3×3 identity is denoted by I₃ (pronounced as "eye-three" or "eye-sub-three", or "the three-by-three identity"). Similarly, the 4×4 identity is I₄ and the 2×2 identity matrix is I₂:

Since identity matrices are, by definition, square matrices, you only need to use one subscript to give their dimensions.

Matrices first saw use in solving systems of linear equations. Related to this origin of matrices, we have two more important matrix types: the augmented matrix and the coefficient matrix.

What is an augmented matrix?

An augmented matrix is derived from a system of linear equations. The augmented matrix contains only the coefficients of the terms in the equations; each column contains the coefficients for one of the original system's variables; the last column contains whatever numerical values went on the other side of the "equals" sign.

What is a coefficient matrix?

A coefficient matrix is derived from a system of linear equations. The coefficient matrix contains only the coefficients of the terms in the equations; each column contains the coefficients for one of the original system's variables; there is *no* final column containing the strictly numerical terms. That is, the "equals [a number]" part of the system is omitted.

To illustrate this, conside the following system of equations:

To create the augmented matrix, I'll first make sure that all the rows and columns line up, by reordering the terms (if necessary), inserting zeroes for whatever terms might be missing, and — at least when I'm just starting out — inserting a 1 in each term without an explicit coefficient. So the above system is turned into the following, more complete and organized, system:

(Note how I moved the x and z terms, so that each column contained only one variable's coefficients.)

Then the augmented matrix is given by:

Sometimes you'll see an augmented matrix with a dashed vertical line. This dashed line serves to emphasize that the matrix came from a system of equations, with the coefficients of the variables to the left of the line. The vertical line separates the coefficients from the "equals [a number]" part:

The coefficient matrix for the linear system is the same as the augmented matrix, except that it's been shorn of its far-right column:

Unless you're taking a linear algebra course, much of the time that you spend with matrices will be related to solving systems of linear equations. While the lead-up to actually applying matrices for this purpose might seem like overkill, believe me: using a matrix to solve a system of equations (or to solve for just one of the variables) is a real time-saver in the long run.