Inverting by row reduction · Linear Algebra

Intuition

To invert a matrix, write the identity beside it and row-reduce the pair. When the left half has become the identity, the right half has become the inverse. If the left half cannot become the identity, there is no inverse.

Writing down every step of tidying a room lets you undo it later. Row-reducing the identity alongside a matrix writes down every step that turns the matrix into the identity, and that record is the inverse.

Gauss–Jordan inversion

Each row operation is multiplication on the left by an elementary matrix: the matrix obtained by doing that operation to the identity. Reducing $[A\mid I]$ applies one product of elementary matrices, $E=E_k\cdots E_1$ , to both halves at once. If the left half reaches $I$ , then $EA=I$ and the right half holds $E$ , which is $A^{-1}$ .

[A\mid I]\ \longrightarrow\ [I\mid A^{-1}]

Elementary matrices

The elementary matrix of $R_2\to R_2-3R_1$ in size is ; multiplying a two-row matrix by it on the left performs that operation.

Why the right half is the inverse

E_k\cdots E_1A=I\implies A^{-1}=E_k\cdots E_1

Name the product of the elementary matrices $E$ . Multiplying $[A\mid I]$ by $E$ acts on the two halves separately and gives $[EA\mid E]$ . When , the matrix is invertible as a product of invertible matrices, so , and then too: undoes from both sides.

Proof steps

E=E_k\cdots E_1

Name the product of the elementary matrices of all the operations used, in order.

E[A\mid I]=[EA\mid E]

Multiplying on the left acts on the two halves separately.

EA=I

The left half has been reduced to the identity.

A=E^{-1}

Each elementary matrix is invertible, so their product is; multiplying $EA=I$ on the left by $E^{-1}$ gives $A=E^{-1}$ .

AE=E^{-1}E=I\implies A^{-1}=E

So $E$ undoes $A$ from both sides, and it is what the right half holds.

Applications

Practice

Operations Are Matrices

Do the operation to the identity, and you get its elementary matrix. Multiplying on the left by it does the operation to any matrix.

R_1\leftrightarrow R_2:\quad \begin{pmatrix}0&1\\1&0\end{pmatrix}

R_2\to R_2-3R_1:\quad \begin{pmatrix}1&0\\-3&1\end{pmatrix}

Try it

Which matrix, multiplied on the left, exchanges the two rows of any $2$ -row matrix?

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What does multiplying a two-row matrix on the left by $\begin{pmatrix}1&0\\-2&1\end{pmatrix}$ do?

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Reduce the Pair

Write $[A\mid I]$ and apply row operations until the left half is $I$ .

\left(\begin{array}{cc|cc}1&2&1&0\\3&7&0&1\end{array}\right)\xrightarrow{R_2-3R_1}\left(\begin{array}{cc|cc}1&2&1&0\\0&1&-3&1\end{array}\right)

\xrightarrow{R_1-2R_2}\left(\begin{array}{cc|cc}1&0&7&-2\\0&1&-3&1\end{array}\right)

Try it

Invert $A=\begin{pmatrix}1&2\\3&7\end{pmatrix}$ by row reducing $[A\mid I]$ . What is the entry in row 2, column 1 of $A^{-1}$ ?

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What is the entry in row 1, column 3 of the inverse of $\begin{pmatrix}1&0&2\\0&1&0\\0&0&1\end{pmatrix}$ ?

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Row reducing $[A\mid I]$ for a $3\times3$ matrix $A$ produces a row of zeros in the left half. What follows?

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While reducing $[A\mid I]$ , column operations may be mixed in with the row operations.

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What is the inverse of $E=\begin{pmatrix}1&0\\5&1\end{pmatrix}$ ?

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An $n\times n$ matrix with $n$ pivots is invertible.

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What You Learned

Each row operation is a left multiplication by an elementary matrix.
Reduce $[A\mid I]$ ; when the left half is $I$ , the right half is $A^{-1}$ .
A zero row on the left means $A$ is not invertible.

Final checkpoint

Try it

What is the entry in row 1, column 2 of the inverse of $\begin{pmatrix}1&1\\0&1\end{pmatrix}$ ?

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Three row operations with elementary matrices $E_1$ , $E_2$ , $E_3$ , in that order, reduce $A$ to $I$ . What is $A^{-1}$ ?

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Every invertible matrix can be written as a product of elementary matrices.

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Completion

Lesson complete

Great work! You now know how to:

write the elementary matrix of a row operation;
invert a matrix by reducing $[A\mid I]$ ;
recognise from the reduction that no inverse exists;
explain why the right half is the inverse.

Mapping the lesson.

Intuition

Gauss–Jordan inversion

Elementary matrices

Why the right half is the inverse

Proof steps

Applications

Practice

Operations Are Matrices

Reduce the Pair

What You Learned

Final checkpoint

Completion

Lesson complete