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### Section 2.3

Arithmetic Operations on Matrices

2.3 Arithmetic Operations on Matrices

- Definition of Matrix
- Column, Row and Square Matrix
- Addition and Subtraction of Matrices
- Multiplying Row Matrix to Column Matrix
- Matrix Multiplication
- Identity Matrix
- Matrix Equation

Definition of Matrix

- A matrix is any rectangular array of numbers and may be of any size.
- The size of a matrix is nxk where n is the number of rows and k is the number of columns.
- The entry aij refers to the number in the ith row and jth column of the matrix.
- Two matrices are equal provided that they have the same size and that all their corresponding entries are equal.

Example Definition of Matrix

- is a 2x3 matrix.
- The entry a1,2 = -1.
- The entry a2,3 = 7.

Column, Row and Square Matrix

- A row matrix or row vector only has one row.
- A column matrix or column vector only has one column.
- A square matrix has the same number of rows as columns.

is a 1x4 matrix and a row matrix.

is a 3x1 matrix and a column matrix.

Example Column, Row & Square Matrix- is a 2x2 matrix and a square matrix.

Addition and Subtraction of Matrices

- The sumA + B of two matrices A and B is defined only if A and B are two matrices of the same size. In this case A + B is the matrix formed by adding the corresponding entries of A and B.
- Two matrices of the same size are subtracted by subtracting corresponding entries.

Example Addition & Subtraction

is not defined.

Multiplying Row Matrix to Column Matrix

- If A is a row matrix and B is a column matrix, then we can form the product AB provided that the two matrices have the same length. The productAB is a 1x1 matrix obtained by multiplying corresponding entries of A and B and then forming the sum.

Example Multiplying Row to Column

is not defined.

Matrix Multiplication

- If A is an mxn matrix and B is an nxq matrix, then we can form the product AB. The productAB is an mxq matrix whose entries are obtained by multiplying the rows of A by the columns of B. The entry in the ith row and jth column of the product AB is formed by multiplying the ith row of A and jth column of B.

Identity Matrix

- The identity matrix In of size n is the nxn square matrix with all zeros except for ones down the upper-left-to-lower-right diagonal.
- Here are the identity matrix of sizes 2 and 3:

Example Identity Matrix

For all nxn matrices A, In A = A In = A.

Matrix Equation

- The matrix form of a system of linear equations is
- AX = B
- where A is the coefficient matrix whose rows correspond to the coefficients of the variables in the equations. X is the column matrix corresponding to the variables in the system. B is the column matrix corresponding to the constants on the right-hand side of the equations.

Example Matrix Equation

Write the following system as a matrix equation

x y constants

Equation 1

Equation 2

Summary Section 2.3 - Part 1

- A matrix of size mxn has m rows and n columns.
- Matrices of the same size can be added (or subtracted) by adding (or subtracting) corresponding elements.

Summary Section 2.3 - Part 2

- The product of an mxn and an nxr matrix is the mxr matrix whose ijth element is obtained by multiplying the ith row of the first matrix by the jth column of the second matrix. (The product of each row and column is calculated as the sum of the products of successive entries.)

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