This unit treats a grid of numbers as a single object you can add, multiply, and use to solve systems of equations. By the end, you should be able to multiply matrices, find 2x2 and 3x3 determinants, and solve linear systems with Cramer's Rule and row reduction.
Read the overview first, then work through each topic note and try its check-yourself question before you look at the answer.
Unit 4 introduces matrices as a tool for organizing numbers and solving systems of linear equations. You'll learn the rules for combining matrices, how to compute a determinant, and how to solve systems with matrix methods instead of substitution.
Honors Algebra II Unit 4 covers matrices and determinants. You learn matrix dimensions and entries. You learn to add, subtract, and scalar-multiply matrices entry by entry, and to multiply matrices row by column. You compute determinants of 2x2 and 3x3 matrices and use them to test whether a matrix is invertible. Then you solve systems of linear equations two ways, with Cramer's Rule and with row operations on an augmented matrix. You also learn to recognize systems with no solution or infinitely many solutions.
You multiply a row of the first matrix by a column of the second, so the inner dimensions have to match. Order matters, and and are usually different. Some products exist in one order and not the other.
A nonzero determinant means the matrix has an inverse and its system has exactly one solution. A determinant of 0 means the matrix is singular. In that case the system has either no solution or infinitely many.
Swapping rows, scaling a row by a nonzero number, and adding a multiple of one row to another all keep the solution set the same. These are the same moves you used in elimination, written in matrix form. The reduced matrix shows you the solution or tells you there isn't a unique one.
Once you write a system this way, the whole unit fits together. Matrix multiplication explains why reproduces the left side of every equation. If is square, its determinant tells you whether there's a unique solution. Row reduction can solve the system, and Cramer's Rule applies when the determinant is nonzero.
A matrix is a rectangular grid of numbers.
A determinant turns a square matrix into a single number.
An augmented matrix stores a system without writing the variables.
A matrix is a rectangular grid of numbers. Its dimensions are written rows × columns, so a matrix has 2 rows and 3 columns. The entry sits in row , column .
Addition, subtraction, and scalar multiplication all work entry by entry. You can only add or subtract matrices with the same dimensions. The product means you multiply every entry of by 3.
Multiplication works differently. For to exist, the number of columns in must equal the number of rows in . If is and is , then is . Each entry of the product comes from a row of and a column of . Multiply the matching entries, then add the results.
Worked example. Let and . The top-left entry of is . Finishing the other entries the same way gives . Reversing the order gives . The two products are different, so order matters.
An identity matrix acts like the number 1. If is , then , using an identity matrix of the right size on each side.
Check yourself. If is and is , does exist, and does ? (Answer: exists and is . is undefined because has 4 columns and has only 3 rows.)
| Operation | Dimension requirement | Size of result |
|---|---|---|
| $A + B$ or $A - B$ | Same dimensions | Same as $A$ and $B$ |
| $cA$ (scalar) | None | Same as $A$ |
| $AB$ | Columns of $A$ = rows of $B$ | Rows of $A$ × columns of $B$ |
A determinant turns a square matrix into a single number. For , the determinant is . For a matrix, you expand along the first row with signs . For example, .
The determinant tells you whether a matrix is invertible. If , then has an inverse and the system has exactly one solution. If , the matrix is singular.
Cramer's Rule solves a square system using determinants when the coefficient determinant is nonzero. Let be the determinant of the coefficient matrix. To get , replace the column with the constants and take the determinant. Find the same way using the column. Then and .
Worked example. Solve and . First, . Putting the constants in the column gives , so . Putting the constants in the column gives , so . Both equations check out, since and .
Check yourself. Is invertible? (Answer: No. Its determinant is , so the matrix is singular.)
An augmented matrix stores a system without writing the variables. Each row is an equation, and the last column holds the constants. List the variables in the same order in every equation, and write 0 for any variable that's missing.
Three row operations keep the solutions the same. You can swap two rows, multiply a row by a nonzero number, or add a multiple of one row to another. Use them to reach row-echelon form and back-substitute, or keep going to reduced row-echelon form and read the answer directly.
Worked example. Solve , , . The augmented matrix is . Replacing with gives the row . Replacing with gives . Swap those two rows, then add 3 times the new to to get , so . Back-substituting, gives , and gives .
If a row reduces to , the system has no solution. If a variable has no pivot and no row is contradictory, the system has infinitely many solutions.
Check yourself. After row reducing an augmented matrix, one row reads , with 4 in the constants column. What does that tell you? (Answer: That row means , which is impossible, so the system is inconsistent and has no solution.)
| Method | Best for | Watch out for |
|---|---|---|
| Row reduction | Any system, including ones with no solution or infinitely many | Arithmetic slips, especially with fractions |
| Cramer's Rule | Small square systems (2x2 or 3x3) with a unique solution | It can't be used when $D = 0$ |
| Inverse matrix, $X = A^{-1}B$ | Square systems, especially with a calculator | $A^{-1}$ must go on the left, and it exists only if $\det(A) \neq 0$ |
| Term | Definition |
|---|---|
| Matrix | A rectangular grid of numbers arranged in rows and columns. |
| Dimensions | The size of a matrix given as rows × columns, so a $2 \times 3$ matrix has 2 rows and 3 columns. |
| Square matrix | A matrix with the same number of rows and columns, the only kind that has a determinant. |
| Identity matrix | A square matrix with 1s on the main diagonal and 0s elsewhere. If $A$ is $m \times n$, then $I_mA = AI_n = A$. |
| Determinant | A single number computed from a square matrix that shows whether the matrix is invertible. |
| Singular matrix | A square matrix whose determinant is 0, so it has no inverse. |
| Inverse matrix | The matrix $A^{-1}$ that satisfies $AA^{-1} = A^{-1}A = I$. |
| Augmented matrix | A matrix that holds a system's coefficients plus a final column of constants. |
| Elementary row operations | Swapping rows, scaling a row by a nonzero number, and adding a multiple of one row to another, none of which change a system's solutions. |
| Cramer's Rule | A method that solves a square system by dividing determinants, $x = D_x / D$, and works only when $D \neq 0$. |
Habits from regular multiplication make it feel like order shouldn't matter. With matrices, and are usually different, and one of them may not even exist. Keep the order exactly as written. When solving with an invertible matrix , multiply by on the left.
Addition works entry by entry, so it's tempting to multiply that way too. Matrix multiplication uses a row of the first matrix and a column of the second, multiplying matching entries and adding them. Check the dimensions first. The inner numbers must match, and the outer numbers give the size of the answer.
In first-row expansion, the middle term is subtracted, so the signs go . Many wrong answers come from adding all three terms. Write the signs above the row before you start, and wrap each 2x2 determinant in parentheses.
If , the formulas and divide by zero, so they tell you nothing about and . A zero determinant means there's no unique solution. Switch to row reduction to find out whether the system has no solution or infinitely many.