Lecture notes · Linear Algebra · Lecture 3

Multiplication and inverse matrices

MIT 18.06 Linear Algebra, lecture 3, explained in plain English

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Part 1 · 0:00–10:00

This part teaches two ways to multiply matrices, by single entries and by whole columns, and shows that both give the same answer.

Key ideas

Each entry of the answer comes from one row of A and one column of B. You multiply matching numbers in pairs and add all the products up. 1:18
The sum for entry C 3 4 is A 3 1 times B 1 4, plus A 3 2 times B 2 4, and so on along the row and down the column. This is written as a sum over k. 3:09
The shapes must match. If A is m by n and B is n by p, the answer C is m by p. The number of columns in A must equal the number of rows in B. 5:01
You can also multiply one column of B at a time. Multiplying A by a column of B gives the matching column of the answer. 7:42
Each column of the answer is a combination of the columns of A. The numbers in the matching column of B tell you how to mix the columns of A. 9:23

Words to know

entry: One single number inside a matrix, found by its row number and column number.
indices: The row and column numbers that label one entry, such as the 3 and 1 in A 3 1.
dot product: A way to combine two lists of numbers: multiply matching pairs, then add the results together.
combination: A mix of several things, where each one is scaled by a number and then all are added together.

Example

Think of a recipe book where each column of ingredients is a different dish. If you know how much of each basic ingredient one batch uses, you can work out the amounts for any mix of batches by adding those amounts together. That is like making each column of the answer as a mix of the columns of A.

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Part 2 · 10:00–20:00

This part shows three more ways to think about matrix multiplication, a special case where the result is a full matrix built from one column and one row, and a way to multiply large matrices by cutting them into blocks.

Key ideas

Each row of the answer is a mix (a combination) of the rows of B. So you can see where each piece of the answer comes from. 10:50
Multiplying a column by a row gives a full-sized matrix, not just a single number. Each column of the result is only a multiple of the first column, and each row is only a multiple of the row. 13:04
The product AB can be built by taking the first column of A times the first row of B, the second column times the second row, and adding them all together. 16:32
Some matrices are very special: all their rows lie on one line, and all their columns lie on one line. Those lines are called the row space and the column space. 17:39
You can also cut matrices into smaller blocks and multiply the blocks as if they were numbers, as long as the sizes match properly. 18:46

Words to know

combination: a mix of things, made by multiplying each one by an amount and adding the results together
shape: how many rows and columns a matrix has, written like 'm by one' for m rows and one column
row space: all the mixes you can make from the rows of a matrix. For the special matrix in the lesson, it is just one line.
block multiplication: cutting a big matrix into smaller square pieces called blocks, then multiplying the pieces the way you would multiply the big matrices

Example

Think of each row of B as a paint color. Multiplying by A is like mixing those paints in different amounts, so each row of the answer is a new color made from the original ones. The answer is always a blend of the starting colors.

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Part 3 · 20:00–30:00

This part explains how to multiply matrices in blocks, what an inverse matrix is, and how to tell when a square matrix has no inverse.

Key ideas

A big matrix can be cut into smaller blocks, and you can multiply the blocks just like single numbers, as long as the shapes fit. Each block in the answer comes from block rows times block columns. 20:20
A square matrix A may have an inverse, written A inverse. Multiplying A by its inverse gives the identity matrix, which is the matrix version of the number 1. 22:06
For square matrices, a left inverse also works as a right inverse. So you only need to find one side to know the inverse works both ways. 23:31
Some matrices have no inverse. These are called singular. The 2 by 2 example has columns that lie on the same line, so no mix of its columns can produce the identity. 27:06
The most useful test is this: if you can find a nonzero vector X where A times X equals zero, then A has no inverse. If an inverse existed, you could multiply both sides by it and force X to be zero, which is impossible. 29:09

Words to know

inverse: A second matrix that, when multiplied with the first, gives back the identity matrix. It works like dividing back to 1 with ordinary numbers.
invertible: A matrix that has an inverse. The lecture also calls these non-singular, and the ones without an inverse are singular.
identity matrix: A square grid of numbers with 1s on the diagonal and 0s everywhere else. Multiplying by it changes nothing, like multiplying by 1.
determinant: A single number worked out from a square matrix. If it is zero, the matrix has no inverse. The lecturer says beginners are not expected to use it yet.

Example

Think of a recipe book where two different ingredients always get mixed in exactly the same ratio. You can never separate them back out to get the original amounts. A matrix whose columns lie on the same line is like that, because its information gets squashed together and cannot be undone.

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Part 4 · 30:00–40:00

This part explains why some matrices have no inverse, and how Gauss-Jordan elimination finds the inverse of one that does.

Key ideas

A matrix has no inverse when some combination of its columns gives the zero column. Such a matrix is called singular. 30:00
If an inverse existed for a singular matrix, multiplying by it would force X to be zero. But X is not zero, so no inverse can exist. 30:21
The lecture uses the matrix with rows 1, 3 and 2, 7 as an invertible example. Its columns point in different directions, so the inverse exists. 32:42
Finding the inverse means solving two systems that share the same matrix A. The right sides are the columns of the identity matrix, which has ones on the diagonal and zeros elsewhere. 34:34
Gauss-Jordan solves both systems at once. It puts the identity on the right side of an augmented matrix, then does elimination until the left side becomes the identity, and the inverse appears on the right. 37:43

Words to know

inverse: A matrix that undoes another matrix. If you multiply a matrix by its inverse, you get the identity back.
singular matrix: A matrix that has no inverse, because some combination of its columns gives zero.
identity matrix: A square grid of numbers with ones along the diagonal and zeros everywhere else. Multiplying by it changes nothing.
augmented matrix: A matrix with extra columns attached on the right, so you can carry the right-hand side along while you work on the left.

Example

Think of an undo button for a sequence of moves. If every move can be reversed, you can get back to where you started, which is like having an inverse. If two different moves end up in the same spot, you can never tell which one happened, so no undo can work, which is like a singular matrix.

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Part 5 · 40:00–46:34

This part shows the Gauss-Jordan method: you write A next to the identity, row-reduce until the left side becomes the identity, and the right side then holds the inverse of A.

Key ideas

Gauss-Jordan elimination starts with A and the identity matrix side by side. Row operations are applied until the left half becomes the identity, and the right half then becomes A inverse. 41:06
You can check an inverse by multiplying it by A. If the result is the identity matrix, the inverse is correct. 41:55
Each elimination step can be written as a matrix. Multiplying all those steps together gives one matrix E, and E times A equals the identity. 43:30
Because E times A is the identity, E must be the inverse of A. When E is applied to the identity on the right, the result is A inverse. 44:59
Gauss-Jordan can also be seen as solving n equations at the same time, with n extra columns added. The answers fill up the columns of A inverse. 46:03

Words to know

Inverse matrix: A matrix that undoes another matrix. When you multiply a matrix by its inverse, you get the identity matrix.
Identity matrix: A square grid of numbers with 1s down the diagonal and 0s everywhere else. Multiplying by it changes nothing, like multiplying by 1.
Pivot: The main number in a row that you use to clear out the numbers below or above it during elimination.
Elimination matrix: A matrix that performs one row operation, such as subtracting a multiple of one row from another, when you multiply it by a matrix.

Example

Think of a set of locks and keys where each step of the process undoes the last one. If the full set of steps turns the original lock into a plain, open state, the combined steps must be the thing that undoes the lock. In the same way, the row steps that turn A into the identity are exactly the steps that build A inverse.

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← Lecture 2: Elimination with matrices · Lecture 9: Independence, basis, and dimension →

Notes written by AI (Claude) from the lecture's official transcript, so check anything important against the lecture. MIT OpenCourseWare · CC BY-NC-SA 4.0 · not affiliated with or endorsed by MIT. These notes are shared under the same license.

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