Math · Precalculus · Concept

Matrix row reduction, determinants and inverses

A matrix is a rectangular array of numbers. Writing a linear system as an augmented matrix lets you solve it with three row operations, none of which changes the solutions: Gaussian elimination reaches row echelon form, and reduced row echelon form shows the answer directly. Row reduction also shows when a system has no solution, or a free variable and infinitely many. For a square matrix, a nonzero determinant means an inverse exists.

Systems as matrices

Each row of an augmented matrix is one equation: the coefficients of x, y and z, then the constant. The vertical bar stands for the equals signs.

[\begin{array}{cccc} 1 & 1 & 1 & 6 \\ 2 & - 1 & 1 & 3 \\ 1 & 2 & - 1 & 2 \end{array}]

Three row operations

Swap two rows; multiply a row by a nonzero number; add a multiple of one row to another. Each matches a step on the equations that keeps every solution, so the final matrix has the same solutions as the first.

R_{2} \to R_{2} - 2 R_{1}

Row echelon form and back-substitution

Gaussian elimination clears the entries below each pivot, the first nonzero entry of a row, working from left to right. The result is a staircase: the last row gives one variable, and substituting upward gives the rest.

Reduced row echelon form

Continuing until every pivot is 1 and is the only nonzero entry in its column gives the reduced row echelon form, or RREF, which is the same whatever operations you choose. When the system has one solution, the left side becomes the identity matrix and the last column is the solution.

No solution or infinitely many

A row that reads 0 = c with c ≠ 0 means the system is inconsistent and has no solution. A column without a pivot belongs to a free variable, and the system has infinitely many solutions, written with a parameter.

Determinants

The determinant of a 2 × 2 matrix is ad − bc. It is the factor by which the matrix scales area, and it is negative when the matrix flips orientation. A 3 × 3 determinant expands along a row into 2 × 2 determinants with alternating signs.

det (\begin{array}{cccccccccc} a & b \\ c & d \end{array}) = a d - b c

Inverses

A square matrix has an inverse exactly when its determinant is not zero. For a 2 × 2 matrix, swap the diagonal entries, change the signs of the other two and divide by the determinant. Then AX = B has the solution X = A⁻¹B.

{(\begin{array}{cccccccccc} a & b \\ c & d \end{array})}^{- 1} = \frac{1}{a d - b c} (\begin{array}{cccccccccc} d & - b \\ - c & a \end{array})

\begin{array}{c} {(\begin{array}{cccccccccc} a & b \\ c & d \end{array})}^{- 1} \\ = \frac{1}{a d - b c} (\begin{array}{cccccccccc} d & - b \\ - c & a \end{array}) \end{array}

Matrices as transformations

A 2 × 2 matrix moves every point of the plane, and its columns are where (1, 0) and (0, 1) land. Multiplying two matrices applies one transformation after the other, and the inverse matrix undoes the transformation, as an inverse function undoes a function.

Common mistakes

Applying a row operation to the coefficients but not the constant: every operation acts on the whole row.
Reading a row of zeros, 0 = 0, as no solution: it only means one equation was redundant.
Computing a 2 × 2 determinant as ad + bc instead of ad − bc.
Looking for the inverse of a matrix whose determinant is 0: it has none.

Key terms

Matrix: A rectangular array of numbers in rows and columns. In a system’s coefficient matrix, each row is an equation and each column belongs to one unknown.
Augmented matrix: A matrix holding a system’s coefficients, with the constants from the right-hand sides added as a last column. That column holds the equations’ values, not another unknown.
Elementary row operation: One of three moves that don’t change a system’s solutions: swap two rows, multiply a row by a nonzero number, or add a multiple of one row to another.
Gaussian elimination: Using elementary row operations to simplify a system into echelon form and solve it. Gauss–Jordan elimination continues until pivots are also cleared above, producing reduced row echelon form.
Pivot: The first nonzero entry in a row, used to clear the entries below it (and, in Gauss–Jordan elimination, above it). In reduced row echelon form every pivot is 1, alone in its column.
Reduced row echelon form: The simplest form row reduction reaches: each row’s first nonzero entry is a 1, alone in its column and to the right of the one above, with any zero rows at the bottom. The solutions can be read straight off it.
Determinant: A number computed from a square matrix; for [[a, b], [c, d]] it is ad − bc. It is the matrix’s area or volume scale factor, and the matrix has an inverse exactly when it isn’t zero.
Matrix inverse: The matrix A⁻¹ that undoes A: AA⁻¹ = A⁻¹A = I. Only square matrices with a nonzero determinant have one.
Identity matrix: A square matrix with 1 on the main diagonal and 0 elsewhere. Multiplying a compatible matrix or vector by it leaves that object unchanged.
Singular matrix: A square matrix with no inverse, which happens exactly when its determinant is 0. A system with a singular coefficient matrix has either no solution or infinitely many.

Work through an example

Solve the system x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2 by row reduction.

Solve a 3 × 3 system with row reduction →

Find the determinant of a 3 × 3 matrix →

Find the inverse of a 2 × 2 matrix →

Recognize infinitely many solutions in RREF →

Sources and scope

Authored study material. Tool results depend on the stated inputs and model assumptions.

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Try in the workspace

Open the example inputs, change a value and keep a useful result on your board.

Row-reduce it in Matrices & linear systems Check the solution in Math Open worked example on a board Matrix Transformations in Math Reference

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