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# Operations-with-Matrices

last edited by 11 years, 7 months ago

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A Matrix is a rectanguar array of numbers

## Basic operations

There are a number of basic operations that can be applied to modify matrices called matrix addition, (scalar) multiplication and transposition. These, together with the matrix multiplication introduced below are the most important operations related to matrices.

Matrices can be set up in equations.

They are equivalent if they have the same order and contain the same elements.

## Subtraction

• The order of the matrices must be the same
• Subtract corresponding elements
• Matrix subtraction is not commutative (neither is subtraction of real numbers)
• Matrix subtraction is not associative (neither is subtraction of real numbers)
• Order of the matrices must be the same
• Any matrix plus the zero matrix is the original matrix

EXAMPLES

## Scalar Multiplication

A scalar is a number, not a matrix.

• The matrix can be any order
• Multiply all elements in the matrix by the scalar
• Scalar multiplication is commutative
• Scalar multiplication is associative

## Zero Matrix

• Matrix of any order
• Consists of all zeros
• Denoted by capital O

## Matrix Multiplication

Am×n × Bn×p = Cm×p

• The number of columns in the first matrix must be equal to the number of rows in the second matrix. That is, the inner dimensions must be the same.
• The order of the product is the number of rows in the first matrix by the number of columns in the second matrix. That is, the dimensions of the product are the outer dimensions.
• Since the number of columns in the first matrix is equal to the number of rows in the second matrix, you can pair up entries.
• Each element in row i from the first matrix is paired up with an element in column j from the second matrix.
• The element in row i, column j, of the product is formed by multiplying these paired elements and summing them.
• Each element in the product is the sum of the products of the elements from row i of the first matrix and column j of the second matrix.
• There will be n products which are summed for each element in the product.

## Equality

Two matrices are equal if and only if

• The order of the matrices are the same
• The corresponding elements of the matrices are the same

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