Solving a System with a Matrix Equation
A system of two linear equations can be written as one matrix equation A·X = B. Multiplying both sides by A's inverse strips away the coefficients and leaves X, the column of solutions.
A system like 2x + y = 7 and 3x - y = 8 is really a statement about how one matrix acts on a point (x, y). Write the coefficients as matrix A = [[2,1],[3,-1]], the unknowns as column X = [x,y], and the right side as B = [7,8]. Then A·X = B. Matrix multiplication undoes nothing by itself, so you need something that cancels A. That something is A inverse, the matrix that turns A back into the identity matrix when multiplied against it.
Multiply both sides on the left by A inverse: A−1·A·X = A−1·B. The left side collapses to X, because A−1·A is the identity matrix, which leaves any column untouched. So X = A−1·B. Once you compute A inverse using the determinant and the swap-and-negate rule for a 2x2 matrix, you multiply it by B and read off x and y directly from the resulting column.
This only works when the determinant of A is nonzero. A zero determinant means A has no inverse, and graphically it means the two lines are parallel or identical, so there is no single point solution to strip out. That link between determinant, invertibility, and the geometry of the lines is what ties this topic to the rest of the system-solving toolkit: elimination, substitution, and graphing all describe the same underlying structure, just with different machinery.
Key terms
- Coefficient matrix
- The matrix formed from the x and y coefficients of a linear system, before the variables or constants are attached.
- Inverse matrix
- The matrix that, multiplied by the original matrix, produces the identity matrix.
- Determinant
- A single number computed from a square matrix, here ad - bc for [[a,b],[c,d]], that tells you whether an inverse exists.
- Identity matrix
- The matrix with 1s on the diagonal and 0s elsewhere that leaves any matrix or column unchanged when multiplied.
- Parallel lines
- Two lines with the same slope and no intersection point, which show up as a zero determinant in the matrix form of their system.
Watch it explained
- How it worksHow does multiplying both sides by the inverse matrix strip the unknowns down to a column?
- Worked exampleWalk me through solving 2x + y = 7 and 3x - y = 8 as a matrix equation.
- Common mistakeWhy isn't the matrix method faster than elimination on a typical two-variable ACT item?
- Ties togetherHow does a zero determinant show up as parallel lines once you graph that same system?
Check yourself
End of topic.