This course introduces matrices as rectangular arrays of numbers and builds them into a tool for solving equations. You will learn the order and the special types of matrices, add, subtract and multiply them, and see why the order of multiplication matters. You will meet the transpose and symmetric and skew-symmetric matrices, then find determinants of 2 by 2 and 3 by 3 matrices using minors, cofactors and the properties of determinants. The course ends with the adjoint and inverse of a matrix and with solving systems of linear equations by the inverse matrix, Cramer's rule and row operations.
Middle Years
Matrices
Matrices from the start: order and types, addition and multiplication, transpose, determinants and their properties, adjoint and inverse, and solving linear equations.
- 9 lessons
Overview
Lessons
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Matrices and their order
What a matrix is, its order (rows by columns), naming entries as a with subscripts i and j, and building a matrix from a rule.
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Types of matrices
Row, column, square, zero, diagonal, scalar, identity and triangular matrices, the trace, and when two matrices are equal.
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Adding, subtracting and scaling matrices
Addition, subtraction and multiplication by a scalar, their laws, and solving simple matrix equations.
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Multiplying matrices
The row by column rule, when a product exists and its order, why AB and BA differ, and powers of a square matrix.
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Transpose, symmetric and skew-symmetric matrices
The transpose and its properties including the reversal law, symmetric and skew-symmetric matrices, and splitting a square matrix into one of each.
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Determinants, minors and cofactors
Determinants of 2 by 2 and 3 by 3 matrices, minors and cofactors, singular matrices, and the area of a triangle as a determinant.
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Properties of determinants
Row and column properties that simplify a determinant before expanding it, with |kA| and |AB| = |A||B|.
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Adjoint and inverse of a matrix
The adjoint as the transposed cofactor matrix, A times adj A = |A| I, the inverse of 2 by 2 and 3 by 3 matrices, and when an inverse exists.
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Solving linear equations with matrices
Writing a system as AX = B and solving it by the inverse matrix, Cramer's rule and the Gauss-Jordan method, with word problems.