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Boundless Algebra
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Chapter 9

Matrices

Book Version 13
By Boundless
Boundless Algebra
Algebra
by Boundless
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Section 1
Introduction to Matrices
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What is a Matrix?

A matrix is a rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns.

Addition and Subtraction; Scalar Multiplication

There are a number of operations that can be applied to modify matrices, such as matrix addition, subtraction, and scalar multiplication.

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Matrix Multiplication

When multiplying matrices, the elements of the rows in the first matrix are multiplied with corresponding columns in the second matrix.

The Identity Matrix

The identity matrix $[I]$ is defined so that $[A][I]=[I][A]=[A]$, i.e. it is the matrix version of multiplying a number by one.

Section 2
Using Matrices to Solve Systems of Equations
Matrix Equations

Matrices can be used to compactly write and work with systems of multiple linear equations.

Matrices and Row Operations

Two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations.

Simplifying Matrices With Row Operations

Using elementary operations, Gaussian elimination reduces matrices to row echelon form.

Section 3
Inverses of Matrices
The Inverse of a Matrix

The inverse of matrix $[A]$ is $[A]^{-1}$, and is defined by the property: $[A][A]^{-1}=[A]^{-1}[A]=[I]$.

Solving Systems of Equations Using Matrix Inverses

A system of equations can be readily solved using the concepts of the inverse matrix and matrix multiplication.

Section 4
Determinants and Cramer's Rule
Determinants of 2-by-2 Square Matrices

The determinant of a square matrix is computed by recursively performing the Laplace expansion to find the determinant of smaller matrices.

Cofactors, Minors, and Further Determinants

The cofactor of an entry $(i,j)$ of a matrix $A$ is the signed minor of that matrix.

Cramer's Rule

Cramer's Rule uses determinants to solve for a solution to the equation $Ax=b$, when $A$ is a square matrix.

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