ACT® · Math: Number and Quantity
A Matrix as a Linear System
A system of linear equations can be stored as a matrix equation AX = B, where A holds the coefficients, X holds the variables, and B holds the constants. An augmented matrix folds A and B into one grid.
A system like 2x + 3y = 12 and x - y = 1 is really just two statements about the same variables. Every equation already has a fixed order: a number times x, plus a number times y, equals a number. Strip away the x's and y's and only the numbers are left doing real work. Lining those numbers up in a grid, row by row, stores the whole system without losing any information, because the position of each number tells you which variable it multiplies and which equation it came from.
To build the coefficient matrix A, take each equation as one row, and read off the coefficients of x then y, in that order, keeping their signs. The variables themselves go into a column matrix X, stacked as x on top and y below. The constants on the right side of the equals signs form another column matrix B. Multiplying A by X and setting that equal to B, written AX = B, reproduces the original system exactly: row one of A times X gives the left side of equation one, and that has to equal the first entry of B.
An augmented matrix skips the separate B column and just tacks the constants onto A as one extra column, usually marked off with a vertical bar. Each row still holds one full equation: x coefficient, y coefficient, constant. This single grid is what later gets row-reduced to solve the system, and it is also what feeds directly into Cramer's rule, where swapping a column of A for the constants and comparing determinants hands you each variable's value.
Key terms
- coefficient matrix
- The grid formed from just the numbers multiplying the variables, one row per equation.
- augmented matrix
- The coefficient matrix with an extra column of constants attached, one per equation.
- matrix equation AX = B
- A system written as coefficient matrix A times variable column X equals constant column B.
- determinant
- A single number computed from a square matrix, used in Cramer's rule to solve for each variable.
Watch it explained
- How it worksHow do the coefficients lift out of two equations and drop into a coefficient matrix?
- How it worksHow does an augmented matrix keep the constants on the right side along for the ride?
- Worked exampleWalk me through writing 2x + 3y = 12 and x - y = 1 as one matrix equation.
- Going furtherWhat happens to the grid when the system carries three variables instead of two?
Check yourself
End of topic.