Matrix Operations and Inverse, Study notes of Mathematics

Download Matrix Operations and Inverse and more Study notes Mathematics in PDF only on Docsity!

2.1 Matrix OperationsMatrix Notation: Two ways to denote

matrix In terms of the^ columns

of In terms of the^ entries

of Main diagonal entries:

Zero Matrix: THEOREM:^ Let

be matrices of the samesize, and let be scalars. Then

Example.^ Compute

where Solution.

If^

how many columns

does^ have?^ 1.^2 2.^3^ 3.^ AB^ is notdefined

Row-Column Rule for Computing AB (alternatemethod) If^ is defined, then

Example.^ Compute

if it is defined.

WARNINGS

Powers of

Example.

Theorem.

2.2 The Inverse of a MatrixDefinition.^ An^

matrix^ is said to be invertible^ if there is an

matrix satisfyingwhere^ is the identity matrix. If it exists,

is called the^ inverse

of^ denoted by

2.2 The Inverse of a MatrixDefinition.^ An^

matrix^ is said to be invertible^ if there is an

matrix satisfyingwhere^ is the identity matrix. If it exists,

is called the^ inverse

of^ denoted by Not all^ matrices are invertible.

A matrix which is not invertible is called a

singular^ matrix. An invertible matrix is called

nonsingular^ matrix.

Theorem.^ If^

is an invertible^

matrix, then for

each^ in^

the equation^

has the unique solution _______________.

Example. Use the inverse of to solve Solution.