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Matrices, Elimination & Inverses

Useful preparation: Two Equations, Two Unknowns

For a long equation, swipe sideways inside its formula box to see every term.

Meaning & notation

A matrix records a linear rule as well as a collection of equations. Each column tells where one coordinate direction goes; multiplying by an input vector combines those columns. Elimination changes the way the equations are written while preserving exactly which inputs produce the requested output. This makes pivots evidence about solvability, rather than merely convenient arithmetic.

Matrix and linear system
An m by n matrix A has m rows and n columns. In Ax=b, the unknown vector x has n entries and the output b has m entries.
Pivot
A pivot is a leading nonzero entry used to eliminate entries in its column. In reduced row echelon form, each pivot is 1 and is the only nonzero entry in its column.
Inverse
For a square matrix A, an inverse is a matrix B satisfying AB=BA=I\displaystyle AB = BA = I. It reverses the transformation for every possible input.

Conditions before calculation

  • Row swaps, multiplication of a row by a nonzero scalar, and adding a multiple of one row to another are reversible; apply them to the entire augmented row.
  • A square n by n matrix is invertible exactly when it has n pivots, equivalently a nonzero determinant. A rectangular matrix has no two-sided inverse.
  • A zero coefficient row with a nonzero right side makes a system inconsistent. A free variable creates infinitely many solutions only when the system is consistent.

Why the method works

1. Preserve the equations

Reversibility explains why row operations preserve the solution set. Scaling by zero is excluded because it can erase a restriction.

2. Read the result

First check for a contradictory row. Then identify pivot variables and free variables; back substitution expresses pivots in terms of any free choices.

3. Verify the input

Substitute the candidate into the original equations. A correct solution must match every component of b, including equations that became redundant during elimination.

Ax=bAx=b

Worked examples

EXAMPLE 1

Solve x+2 y=5\displaystyle x + 2\, y = 5 and 3 x+5 y=13\displaystyle 3\, x + 5\, y = 13.

  1. Write coefficients and constants in matching row order.

    [1253513]\left[\begin{array}{cc|c}1&2&5\\3&5&13\end{array}\right]

  2. Subtract three times row one from row two; the x coefficient cancels.

    R2←R2−3R1:−y=−2R_2\leftarrow R_2-3R_1:\quad -y=-2

  3. Thus y=2\displaystyle y = 2 and the first equation gives x=1\displaystyle x = 1. Substitution checks both equations.

    1+2(2)=5,3(1)+5(2)=131+2(2)=5,\qquad 3(1)+5(2)=13

Conclusion: The unique solution is (1,2); two pivots leave no free variable.

EXAMPLE 2

Classify x+2 y=3\displaystyle x + 2\, y = 3 and 2 x+4 y=6\displaystyle 2\, x + 4\, y = 6, then change the last constant to 7.

  1. Eliminate the second x coefficient; with constant 6, the entire second row becomes zero.

  2. Choose y=t\displaystyle y = t freely and recover x from the first row.

    (x,y)=(3−2t,t),t∈R(x,y)=(3-2t,t),\quad t\in\mathbb R

  3. With constant 7, the same operation produces a contradiction.

    0x+0y=10x+0y=1

Conclusion: The first system has infinitely many solutions; the altered system has none. The shared coefficient matrix is singular.

Pitfalls & counterexamples

Tempting claim: A zero determinant means the system has no solution.

Correction: It means the coefficient matrix is not invertible; the right side determines whether solutions are absent or nonunique.

Test the claim: The zero matrix with zero right side accepts every input vector.

Check your understanding

Can swapping two equations change the solution set?

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Compare your reasoning

No.

Both restrictions are still imposed, and swapping them back reverses the operation.

Connect this idea

Elimination reveals rank and nullity. In computation, solving by pivoted elimination is generally preferable to explicitly forming an inverse; small pivots also signal the need to consider numerical accuracy.

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Reference reading

Use these texts to study the underlying definitions and theorems in greater depth. The explanations and examples above are written for Numerive.

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