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Chapter 2 Matrices Definition of a matrix

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A system of 3 equations: Represented by a matrix:

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**Types of Matrices Square matrix: # of rows = # of columns**

upper triangular matrix strictly upper triangular matrix

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**lower triangular matrix strictly lower triangular matrix**

diagonal matrix

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banded matrix a square matrix with elements of zero except for the principal diagonal and values in the positions adjacent to the diagonal. tridiagonal matrix

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**unit matrix: 1 on the principal diagonal**

null matrix: All elements are zero.

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symmetric matrix: a square matrix in which skew-symmetric matrix: a square matrix in which for all i and j

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**transpose of matrix A: AT**

(AT) T = A

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**Matrix Operations Matrix equality Matrix addition and subtraction**

C = A + B = B + A (commutative) C = A - B

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**Example: Matrix addition and subtraction**

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**Matrix Multiplication**

One example

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**Rules of Matrix Multiplication**

# of columns in A = # of rows in B # of rows in C = # of rows in A # of columns in C = # of columns in B

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**5. Matrix multiplication is not commutative**

6. Matrix multiplication is associative

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**Example: Matrix Multiplication**

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**Matrix Multiplication by a Scalar**

An example:

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Matrix Inversion where A-1 is the inverse of A, and I is the unit matrix

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**Example: Matrix Inversion**

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Matrix Singularity If the inverse of a matrix A exists, then A is said to be nonsingular. If the inverse of a matrix A does not exist, then A is said to be singular. If matrix A is singular, then the linear system of simultaneous equations represented by A has no unique solution.

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**There are an infinite number of solutions if 2a = b.**

There is no feasible solution if 2a b. Thus matrix A is singular.

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**trace of a square matrix = sum of diagonal elements**

matrix augmentation: addition of a column or columns to the initial matrix

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matrix partition

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Vectors Column vector Row vector Vectors of two ordinates

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orthogonal vectors Two vectors are said to be orthogonal if their product is equal to zero. If two vector are orthogonal, they are perpendicular to each other in the n-dimensional space.

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normalized vectors A vector is normalized by dividing each element by its length. A normalized vector has a length 1. Two vectors that are both normalized and orthogonal to each other are said to be orthonormal vectors.

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Example: Vectors

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Normalized vectors:

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**Determinants A determinant of a matrix A is denoted by |A|.**

The determinant of a 22 matrix: The determinant of a 33 matrix:

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**The determinant of an nn matrix: **

The minor of aij, denoted by Aij, is the matrix after removing row i and column j. The determinant of an nn matrix: The general expression for the determinant of an nn matrix:

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**Example: Matrix Determinant**

with the first row and their minors:

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**with the second column and their minors:**

Since |A|=0, A is a singular matrix; that is the inverse of A doest not exist.

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**Properties of Determinants**

1. If the values in any row (column) are proportional to the corresponding values in another row(column), the determinant equals zero

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**2. If all the elements in any row(column) equal zero, the determinant equals zero.**

3. If all the elements of any row(column) are multiplied by a constant c, the value of the determinant is multiplied by c.

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4. The value of the determinant is not changed by adding any row (column) multiplied by a constant c to another row (column). 5. If any two rows (columns) are interchanged, the sign of the determinant is changed.

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**6. The determinant of a matrix equals that of its transpose; that is, |A| = |AT|.**

7. If a matrix A is placed in diagonal form, then the product of the elements on the diagonal equals the determinant of A.

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8. If a matrix A has a zero determinant, then A is a singular matrix; that is, the inverse of A does not exist.

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Rank of A Matrix A matrix of r rows and c columns is said to be of order r by c. If it is a square matrix, r by r, then the matrix is of order r. The rank of a matrix equals the order of highest-order nonsingular submatrix.

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**Example 1: Rank of Matrix**

3 square submatrices: Each of these has a determinant of 0, so the rank is less than 2. Thus the rank of R is 1.

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**Example 2: Rank of Matrix**

Since |A|=0, the rank is not 3. The following submatrix has a nonzero determinant: Thus, the rank of A is 2.

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