Matrices

1
Matrices
Matrix - Ordered set of scalars arranged in a
rectangular pattern

• matrix with 3 rows and 3 columns
• matrix is of order 3x3
• matrix is a square matrix of order 3

• matrix with 3 rows and 2 columns
• matrix is of order 3x2

2

• Create an m-file with the code shown.
• Elements of a matrix are entered row-by-row with
  a semicolon separating the rows.
• Inclusion of a carriage return between rows is
  optional.
• Save the file in your workspace as matrix_1.m

• Execute the program from the Command Window and
  verify the results shown.

3

• A matrix with a single row is referred to as a
  row vector
• A matrix with a single column is referred to as a
  column vector
• A matrix with a single number can still be
  treated as a matrix. It is referred to as a
  scalar
• An individual element in a matrix is identified
  by its row and column number

4

• Add the code shown to the end of your matrix_1.m
  file.
• Save the changes, execute the file from the
  Command Window and verify the additional output
  shown.
• Individual elements of a matrix are identified by
  row and column numbers in parenthesis by the
  matrix name.
• Elements of a matrix can be changed individually.
• Example of a row vector.
• Example of a column vector.

5
Scalar Multiplication

• In the scalar multiplication of a matrix, each
  element of the matrix is multiplied by the same
  scalar value

6
Matrix Addition or Subtraction

• For matrix addition (or subtraction), the
  elements of two matrices are added (or
  subtracted) on element by element basis.
• The two matrices must be of equal order, i.e.,
  they must have the same number of rows and
  columns.
• The result will be of the same order.

7

• Modify matrix_1.m by adding the lines shown.
• Save the changes, execute the program, and verify
  the additional output shown.
• Scalar multiplication of a matrix.
• Matrix addition.

8
Matrix Multiplication

• For matrix multiplication, the elements in the
  rows of the first matrix are multiplied by
  elements of the columns of the second matrix and
  the products are added.
• The matrices must have opposing order, i.e., the
  number of columns of the first matrix must equal
  the number of rows of the second matrix.

• Result is a square matrix

9

• Modify matrix_1.m by adding the lines shown.
• Save the changes, execute the program and verify
  the additional output shown.
• Matrix multiplication of a 2x3 by a 3x3
  results in a 2x3 matrix
• Matrix multiplication of a 3x3 by a 2x3 is
  invalid and results in an error message.
• Matrix multiplication requires matrices of
  opposing order p x mm x q
  p x q

10
Special Matrices

• For the transpose of a matrix, the rows and
  columns are exchanged.

• The identity matrix has 1s on the diagonal and
  0s for all of the off-diagonal elements.
• Multiplying a matrix by the identity matrix
  yields the same original matrix.
• Multiplying by the identity matrix is analogous
  to multiplying a scalar by one

11
Special Matrices

• Multiplying a matrix by its inverse results in
  the identity matrix.
• Multiplying a matrix by its inverse is analogous
  to dividing a scalar by itself the result is
  one.

• The process for the determination of the inverse
  of a matrix is well defined but tedious. Tools
  are available to do this in Excel, Matlab, and
  Mathcad.

12

• Modify matrix-1.m by adding the lines shown.
• Note that to avoid generating the matrix multiply
  error, the incorrect line has been turned into an
  non-executable comment by beginning the line with
  a character.
• Save the changes, execute the program, and verify
  the additional output shown.
• In Matlab, the transpose of a matrix is indicated
  by adding an apostrophe after the matrix name.

13

• Modify matrix-1.m by adding the lines shown.
• Save the changes, execute the program, and verify
  the additional output shown.
• The identity matrix has 1s on the diagonal and
  0s for all other elements.
• Multiplying a matrix by the identity matrix
  yields the original matrix.

14
Solution of Linear Equation Sets

• Linearly independent equation set

15
Solution of Linear Equation Sets

• Multiply through by the inverse of the
  coefficient matrix
• The left-hand side is just the vector of unknowns
• Evaluating the right-hand side produces the
  solution

16
Solution of Linear Equation Sets

• Perform a check by multiplying the original
  coefficient matrix by the solution vector.
• The result should be equal to the right-hand side
  vector from the equation set.

17

• Modify matrix_1.m by adding the lines shown.
• Save the changes, execute the program and verify
  the additional output shown.
• Matlab includes a built-in function for
  evaluating the inverse of a matrix, inv().
• Note that multiplying a matrix by its inverse
  results in the identity matrix.
• Calculate the solution for the linear equation
  set by multiplying the right-hand-side vector by
  the inverse of the coefficient matrix.