Matrices

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.

Row-echelon form and Gaussian elimination

Learn how to use the matrix row operations in order to easily solve large systems of linear equations.
7:37
Solving linear systems with matrices
Sal solves a linear system with 3 variables by representing it with an augmented matrix and bringing the matrix to reduced row-echelon form.

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!).
Article
Matrix multiplication dimensions
Learn about the conditions for matrix multiplication to be defined, and about the dimensions of the product of two matrices.
3:32
Defined matrix operations
Sal discusses the conditions of matrix dimensions for which addition or multiplication are defined.
Article
Intro to identity matrices
Learn what an identity matrix is and about its role in matrix multiplication.
7:59
Intro to identity matrix
Just as any number remains the same when multiplied by 1, any matrix remains the same when multiplied by the identity matrix. Learn more from Sal.
3:44
Dimensions of identity matrix
Sal explains why the identity matrix is always a square matrix, even though it works with non-square matrices.
Article
Properties of matrix multiplication
Learn about the properties of matrix multiplication (like the distributive property) and how they relate to real number addition.
7:32
Is matrix multiplication commutative?
Sal checks whether the commutative property applies for matrix multiplication. In other words, he checks whether for any two matrices A and B, A*B=B*A (the answer is NO, by the way).
8:37
Associative property of matrix multiplication
Sal shows that matrix multiplication is associative. Mathematically, this means that for any three matrices A, B, and C, (A*B)*C=A*(B*C).
4:42
Zero matrix & matrix multiplication
Just as any number multiplied by zero is zero, there is a zero matrix such that any matrix multiplied by it results in that zero matrix. Learn more from Sal.
3:30
Using properties of matrix operations
Sal determines which of a few optional matrix expressions is equivalent to the matrix expression A*B*C. This is done using what we know about the properties of matrix addition and multiplication.
3:58
Using identity & zero matrices
Sal solves a problem where he has to determine whether unknown matrices are zero or identity to make an equation true.

The determinant of a 2x2 matrix

Learn what the determinant of a matrix is and how to find it for 2x2 matrices.
1:10
Determinant of a 2x2 matrix
Sal shows how to find the determinant of a 2x2 matrix.
Exercise
Determinant of a 2x2 matrix
Find the determinant of a given 2x2 matrix.

Finding the inverse of a matrix using its determinant

Learn about a way to find the inverse of a matrix using its determinant and adjuagates.
2:48
Finding inverse of a 2x2 matrix using determinant & adjugate
Sal gives an example of how to find the inverse of a given 2x2 matrix.

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.
Exercise
Find the inverse of a 2x2 matrix
Find the inverse matrix of a given 2x2 matrix.