Augmented Matrices

Linear Algebra (Matrices) · Axiom Academy

The single object that packs a whole linear system into a grid of numbers — coefficients on the left, constants on the right, one row per equation. 1. Packing a System Into a Grid Start with three equations in three unknowns. Watch each equation's coefficients slide left into the matrix A , while its constant flies past the bar into the column b . The variable symbols just fall away — their position in the grid records which variable each number belonged to. A system of three linear equations 2. Coefficient Matrix vs. Augmented Matrix The coefficient matrix A holds only the numbers multiplying the variables. By itself it has forgotten the right-hand sides. To recover the full system we augment it: drop in a vertical bar and append the constants as one extra column. The animation builds A first, then slides the bar and the b -column into place. One equation of the system — read left to right across the row. One variable's coefficients — column 1 is the x 's, column 2 the y 's, column 3 the z 's. The constants b — the right-hand sides, sitting past the bar. A bookkeeping mark: it remembers where every " = " used to be. 3. Reading a Row Back as an Equation Each row of is one equation in disguise. Highlight a row, multiply its first three entries by x , y , z , set the entry past the bar as the right-hand side — and the original equation reappears. Below, row 2 is lit and its numbers fly out to rebuild x - 2y + 3z = 1 .

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.