Matrices

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This course is a part of Precalculus, a 8-course Topic series from Khan Academy.

Learn what matrices are and about their various uses: solving systems of equations, transforming shapes and vectors, and representing real-world situations. Learn how to add, subtract, and multiply matrices, and find the inverses of matrices.

This course contains 16 segments:

Introduction to matrices

Learn what matrices are and about their parts (coordinates and dimensions).

Representing linear systems of equations with augmented matrices

Learn how systems of linear equations with multiple variables are represented with matrices.

Elementary matrix row operations

Learn about the three basic operations we can perform on a matrix without changing the solution set of the linear system it represents.

Row-echelon form and Gaussian elimination

Learn how to use the matrix row operations in order to easily solve large systems of linear equations.

Adding and subtracting matrices

Learn how to add and subtract two matrices to obtain a new matrix (just like numbers!).

Multiplying matrices by scalars

Learn how to multiply matrices by scalars (in the world of matrices, scalars are simply regular numbers). For example, multiply a 2X2 matrix by 3.

Properties of matrix addition & scalar multiplication

Learn about zero matrices and about how matrix addition, subtraction, and scalar multiplication are related to their corresponding operations with real numbers.

Multiplying matrices by matrices

Learn how to find the result of a multiplication of two matrices.

Properties of matrix multiplication

Learn about identity matrices and about how matrix multiplication relates to real number multiplication (spoiler: the distributive property applies but the commutative property does not!).

Matrices as transformations

Learn how matrices can be used as transformations of vectors or transformations of the plane.

The determinant of a 2x2 matrix

Learn what the determinant of a matrix is and how to find it for 2x2 matrices.

Introduction to matrix inverses

Learn what the inverse of a matrix is, and how to determine whether two matrices are inverses or whether a matrix is invertible.

Finding the inverse of a matrix using its determinant

Learn about a way to find the inverse of a matrix using its determinant and adjugates.

Practice finding the inverses of 2x2 matrices

After you've learned how to find the inverse of a 2x2 matrix, gain some practice with it.

Solving equations with inverse matrices

Learn how to solve systems of linear equations using inverse matrices, and how to solve matrix equations using inverse matrices.

Model real-world situations with matrices

Learn how matrices can be used to describe real-world situations.