Lecture notes · Linear Algebra · Lecture 2

Elimination with matrices

MIT 18.06 Linear Algebra, lecture 2, explained in plain English

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Part 1 · 0:00–10:00

This part introduces elimination, the step-by-step method for solving a system of equations by turning the matrix into a simpler triangular form.

Key ideas

Elimination is the method used by every software package to solve equations, and it works well when the matrix is a good one. It is used here instead of determinants. 0:47
The first step is to keep the first equation, multiply it by a chosen number, and subtract it from the second equation so that the x term in the second equation disappears. 3:26
The number you start with in each step is called the pivot, and the number you multiply by is called the multiplier. The multiplier is picked so the unwanted term cancels out. 4:10
Pivots can never be zero. If a pivot were zero, the method would fail. 9:47
The goal of elimination is to turn the original matrix A into an upper triangular matrix U, which has zeros below the main diagonal and shows the three pivots clearly. 9:24

Words to know

elimination: A method of solving equations by removing one unknown at a time, using subtraction, until the answer is easy to find.
pivot: The key number you start with in a step. It is the number you use to clear out the numbers below it.
multiplier: The number you multiply an equation by before subtracting it, chosen so that a chosen term cancels to zero.
upper triangular: A matrix with zeros in every position below the main diagonal, so the non-zero numbers form a triangle in the upper part.

Example

Think of a set of clues where you want to find three hidden numbers. You use one clue to remove one unknown from another clue, then keep going until only one unknown is left. Elimination works the same way, removing one variable at a time so the answer becomes easy to read off.

3 checked practice questions for this part are in the app.

Part 2 · 10:00–20:00

This part explains when elimination can fail, how a row swap can rescue it, how the right-hand side is carried along, and how back substitution solves the final triangular equations.

Key ideas

A pivot is the number used to clear out the numbers below it. If a 0 lands in the pivot spot, the method is in trouble, but it is not always a dead end. 11:00
If the pivot spot holds a 0, you can swap rows to bring a nonzero number up. You only fail completely when there is a 0 in the pivot spot and no nonzero number below it to swap in. 11:21
The right-hand side numbers (the b column) are added as an extra column called the augmented matrix. Every step done to the left side is also done to the right side, so the equations stay true. 14:46
After elimination, the equations are triangular, so you solve the last one first and then work upward. This is called back substitution. 17:59
The lecturer says the elimination steps can also be written as matrices. He plans to show this next, starting with the first step. 19:04

Words to know

pivot: The number you use as the main anchor when you clear out the numbers below it in a column.
row exchange: Swapping two equations (rows) in a matrix, usually to move a nonzero number into the pivot spot.
augmented matrix: A matrix with the right-hand side numbers tacked on as an extra column, so the equations and their answers are kept together.
back substitution: Solving the last equation first, then plugging that answer into the equation above it, and repeating until you reach the top.

Example

Think of a set of stacked bricks where each brick must be placed on top of the one below it. If the spot you need is empty, you might swap in a different brick from below to fill it. Only if no bricks are left to swap in do you truly get stuck. Back substitution is like building from the top of the stack down to the bottom: you know the top piece first, and each piece below it then becomes easier to figure out.

3 checked practice questions for this part are in the app.

Part 3 · 20:00–30:00

This part shows how a row operation, like subtracting 3 times row one from row two, can be written as a matrix multiplication.

Key ideas

Multiplying a matrix by a column vector gives a combination of the matrix's columns. This is the big picture the lecturer wants you to keep in mind. 21:02
Multiplying a row vector by a matrix gives a combination of the matrix's rows. Elimination uses row operations, so this view matters for this course. 23:41
The identity matrix does nothing when you multiply by it. It works like the number 1 does for ordinary numbers. 26:40
To subtract 3 times row one from row two, you use a simple matrix. Its top row is 1 0 0, its middle row is -3 1 0, and its bottom row is 0 0 1. 27:24
You can check one entry of a product by taking one row from the first matrix and one column from the second, multiplying matching numbers, and adding them up. The lecturer calls this simple matrix an elementary matrix, labeled E21. 28:52

Words to know

Linear combination: A mix of things where each one is multiplied by a number and then added together, like 1 times row one plus 2 times row two.
Identity matrix: A matrix with 1s on the diagonal and 0s elsewhere. Multiplying by it leaves everything unchanged.
Elementary matrix: A simple matrix that performs one basic step of elimination, such as subtracting a multiple of one row from another.
Dot product: A way to get one number by multiplying matching entries of a row and a column, then adding those results.

Example

Think of a recipe where each row is an ingredient list. Taking 3 cups of flour away from a second recipe card is like changing one row using another row. Writing that one change as a single simple grid is what the elementary matrix does.

3 checked practice questions for this part are in the app.

Part 4 · 30:00–40:00

This part shows how a row-by-row cleanup of a matrix can be packed into one single matrix, and how row swaps are handled with a permutation matrix.

Key ideas

Each elimination step can be written as its own matrix, called an elimination matrix. For example, E32 fixes the position in row 3, column 2 by subtracting 2 times row 2 from row 3. 30:23
Doing the steps one after another, E32 times E21 times A, gives the final result U. Those steps can be combined into one matrix that does all of elimination in one shot. 32:46
You can change the grouping of the multiplications (the parentheses) but not the order of the matrices. This is the associative law, and it is very important in linear algebra. 34:41
A permutation matrix exchanges rows. You can find it by exchanging the same rows of the identity matrix. 38:26

Words to know

elimination matrix: A matrix that performs one clearing step on another matrix, such as subtracting a multiple of one row from another row.
pivot: The number in the position you are clearing from, which is used as the base for the subtraction step.
associative law: A rule that says you can move the parentheses in a multiplication, as long as you keep the matrices in the same order.
permutation matrix: A matrix that swaps rows (or columns) of another matrix when you multiply by it.

Example

Think of a recipe made of several steps, written as a list of small instructions. You can follow the steps one by one, or you can combine them into one shorter set of instructions that gives the same finished dish. The lesson works the same way: many small elimination steps can be combined into one matrix that gives the same result.

3 checked practice questions for this part are in the app.

Part 5 · 40:00–47:23

This part explains that a matrix placed on the left of another matrix does row operations, that the order of matrices matters, and that an inverse matrix undoes a step.

Key ideas

To do a column operation, the matrix goes on the right. To do a row operation, the matrix goes on the left. The lecturer says row operations are what we really do. 40:22
Matrices cannot be swapped around. A times B is not the same as B times A, so you must keep them in the order they were written. 41:30
Instead of multiplying all the elimination steps together, it is easier to ask how to get from U back to A by reversing the steps. 43:00
An inverse matrix undoes a step. If a step subtracted 3 times row 1 from row 2, the inverse step adds 3 times row 1 back to row 2. 45:42
The inverse of a matrix E is written as E to the minus one, or E inverse. Multiplying E by its inverse gives the identity matrix. 47:08

Words to know

Inverse: A matrix that undoes what another matrix did. Multiplying a matrix by its inverse gives back 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.
Permutation matrix: A matrix made by rearranging the rows or columns of the identity matrix. It can swap rows or columns.
Commutative law: The rule that you can swap the order of numbers in a multiplication and get the same answer. For matrices, this rule does not hold.

Example

Think of putting on a sock, then undoing it by taking the sock off. Taking the sock off is the inverse step, because it reverses what you did. In the lesson, adding 3 times row 1 back to row 2 reverses the earlier step of subtracting it.

3 checked practice questions for this part are in the app.

← Lecture 1: The geometry of linear equations · Lecture 3: Multiplication and inverse matrices →

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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