Writing a system as a matrix equation

When you reach for it

A linear system such as

carries three kinds of information: the numbers multiplying the variables, the variables themselves, and the numbers on the right. Elimination copies the letters x and y over and over, although only the numbers ever change.

Writing the system as a matrix equation AX = B puts the three kinds of information in three matrices and packs the whole system into one statement. You reach for it whenever a system is to be handled with matrices: by row operations, an inverse matrix, Cramer’s rule, or a calculator.

The procedure

The three matrices

Take a system of m linear equations in n variables, each equation written with the variable terms on the left, in the same order in every equation, and the constant on the right.

• The coefficient matrix A is m \times n. Row i holds the coefficients of equation i; column j holds every coefficient of variable j. A variable that does not appear in an equation has coefficient 0 there. A variable with no number written in front, as in x or -y, has coefficient 1 or -1.
• The variable matrix X is the n \times 1 column of variables, in the chosen order.
• The constant matrix B is the m \times 1 column of right-hand sides.

For the system above,

and the system becomes

Why the one equation is the whole system

The product AX is (m \times n)(n \times 1): the inner sizes match, so by the row-by-column rule it exists and is m \times 1, the same size as B. Its entry in row i is row i of A times the column X: the first coefficient times the first variable, plus the second times the second, and so on. That sum is exactly the left side of equation i. Here

Two matrices are equal when they have the same size and every pair of matching entries is equal (as in Rows, columns and dimensions). So AX = B says 5x + 2y = 12 and 3x - 4y = 2: one equation between columns is m equations between numbers. It follows that a column X of numbers makes AX = B true exactly when those numbers solve every equation of the system. The matrix equation and the system have the same solutions, because they are the same statement.

Two habits follow from the argument. Line every equation up first, since row i of A must be read against the same list of variables as every other row; a coefficient in the wrong column multiplies the wrong variable. And to check the work, multiply AX back out: the rows must reproduce the original equations.

Read the other way, any matrix equation AX = B with X a column of variables can be read back as a system: each row of A, paired with the matching entry of B, is one equation.

Worked examples

Boundaries

• Linear equations only. Each entry of AX is a sum of constants times single variables. An equation with xy, x^2, \displaystyle \frac{1}{x} or \sin x has a term of no such form and cannot be written this way.
• Coefficients are constants. An entry of A cannot contain a variable of X; a term such as x \cdot y is not “coefficient x times y”.
• One variable order for every row. The columns of A and the entries of X follow a single order. Writing one row in a different order produces a matrix equation for a different system.
• A comes first. The product XA is (n \times 1)(m \times n), which is undefined unless m = 1, and even then it is not the system. Matrix multiplication is not commutative.
• Writing is not solving. Every linear system can be written as AX = B, whether it has one solution, none or infinitely many. The matrix equation has the same solutions as the system and does not decide which case holds.

Where this goes next

Once a system is AX = B, the letters can be dropped altogether. Setting B beside A gives the augmented matrix of Augmented matrices and row operations, where elimination becomes operations on rows. When A is square and has an inverse, multiplying both sides by it gives X = A^{-1}B, the method of Inverse of a 2 by 2 matrix, and solving with it; whether that inverse exists is decided by the determinant, and Cramer's rule writes each variable as a ratio of determinants. Read column by column instead of row by row, AX is a combination of the columns of A, so solving the system asks which combination of those columns makes B.

Exercises

Write the system as a matrix equation

1. 3x + 7y = 1, 2x - y = 9
2. x - y = 0, 6x + 5y = -8
3. 2x + y - 3z = 8, x - 4y + z = -2, 5x + 2y + 6z = 0
4. p + q = 6, 2q - r = 1, 3p + 5r = -4
5. y = 2x + 5, 3x = 4 - y
6. \displaystyle \frac{x}{2} + \frac{y}{3} = 1, \displaystyle \frac{2}{5}x - y = 2
7. z = x + y, 2x + z = 7 + y, x - 3 = 2y

Read it back as a system

8. \begin{bmatrix} 4 & -3 \\ 1 & 6 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 10 \\ -5 \end{bmatrix}
9. \begin{bmatrix} 1 & 0 & -2 \\ 0 & 3 & 1 \\ -1 & 1 & 0 \end{bmatrix}\begin{bmatrix} u \\ v \\ w \end{bmatrix} = \begin{bmatrix} 5 \\ 0 \\ 2 \end{bmatrix}
10. \begin{bmatrix} 2 & -1 & 3 \\ 0 & 1 & -4 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 6 \end{bmatrix}. How many equations and how many variables does the system have?

Test a column

11. Write 2x + 5y = 1, x - 3y = 6 as AX = B. Decide whether (3, -1) and (-2, 1) are solutions by computing AX.
12. Write x + 2y + z = 1, 2x - z = -2, y + 3z = 10 as AX = B, and decide whether (1, -2, 4) is a solution.
13. The column \begin{bmatrix} 2 \\ 5 \end{bmatrix} solves \begin{bmatrix} k & 2 \\ 3 & -1 \end{bmatrix}X = \begin{bmatrix} 16 \\ 1 \end{bmatrix}. Find k, and check the other row.

In the woodshop

14. Nabil sells cutting boards at 18 dollars and stools at 45 dollars. He sells 20 items and takes in 576 dollars. Write the conditions as AX = B with X = \begin{bmatrix} c \\ s \end{bmatrix}, and check that 12 cutting boards and 8 stools fit.
15. Samir builds small, medium and large planter boxes, selling at 12, 20 and 30 dollars. He sells 30 boxes for 544 dollars, and sells three times as many small boxes as large ones. Write AX = B with X = \begin{bmatrix} s \\ m \\ l \end{bmatrix}, and check that 12 small, 14 medium and 4 large fit.
16. Layla’s plan for cutting boards b and shelves h is written as \begin{bmatrix} 1 & 1 \\ 2 & 4 \end{bmatrix}\begin{bmatrix} b \\ h \end{bmatrix} = \begin{bmatrix} 25 \\ 68 \end{bmatrix}, where row 1 counts items and row 2 counts board feet of lumber. Write the two equations, say what each one states, and check whether 16 cutting boards and 9 shelves fit.

Reasoning

17. A system has 3 equations in 4 variables. Give the sizes of A, X and B, and explain each from the row-by-column rule.
18. In the system 2x + y = 5, y - x = 1, the second equation lists y before x. Write the system as a matrix equation with X = \begin{bmatrix} x \\ y \end{bmatrix}. Then explain why \begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 1 \end{bmatrix} states a different system.
19. Samir multiplies the second equation of Example 1 by 3 before writing the matrix equation. His A and B differ from those in Example 1. Explain why his matrix equation still has exactly the same solutions.
20. Explain why \begin{bmatrix} 0 & 0 \\ 1 & 2 \end{bmatrix}X = \begin{bmatrix} 3 \\ 1 \end{bmatrix} has no solution. What changes if the 3 is replaced by 0?