Matrices and Determinants

Matrices A matrix is a rectangular arrangement

of numbers in rows and columns. Rows run horizontally and columns run vertically.

The dimensions of a matrix are stated “m x n” where ‘m’ is the number of rows and ‘n’ is the number of columns.

Equal Matrices

Two matrices are considered equal if they have the same number of rows and columns (the same dimensions) AND all their corresponding elements are exactly the same.

Types of Matrices

1. Rectangular Matrix2. Square Matrix3. Diagonal Matrix4. Scalar Matrix5. Identity Matrix6. Null Matrix

7. Row Matrix8. Column Matrix9. Upper Triangular

Matrix10. Lower Triangular

Matrix11. Sub matrix.

Matrix Addition

You can add or subtract matrices if they have the same dimensions (same number of rows and columns).

To do this, you add (or subtract) the corresponding numbers (numbers in the same positions).

Matrix Addition

2 4 1 05 0 2 11 3 3 3

Example:

3 47 12 0

Properties of Matrix Addition

Matrix addition is commutative i.e. A+B = B+A

Matrix addition is associative i.e. (A+B)+C = A+(B+C)

Matrix addition is distributive w.r.t. scalar K K(A+B) = KA+KB

Scalar Multiplication To do this, multiply each entry in

the matrix by the number outside (called the scalar). This is like distributing a number to a polynomial.

Scalar Multiplication

2 44 5 0

1 3

Example:

8 1620 04 12

Matrix Multiplication

Matrix Multiplication is NOT Commutative! Order matters!

You can multiply matrices only if the number of columns in the first matrix equals the number of rows in the second matrix.

2 3 5 69 7

2 columns2 rows

1 2 03 4 5

Matrix Multiplication

Take the numbers in the first row of matrix #1. Multiply each number by its corresponding number in the first column of matrix #2. Total these products.

2 3 5 69 7

1 2 03 4 5

21 33 11

The result, 11, goes in row 1, column 1 of the answer. Repeat with row 1, column 2; row 1 column 3; row 2, column 1; ...

Matrix Multiplication

Notice the dimensions of the matrices and their product.

2 3 5 69 7

1 2 03 4 5

11 8 1513 34 30 12 46 35

3 x 2 2 x 3 3 x 3__ __ __ __

Matrix Multiplication Another example:

2 15

9 02

10 5

3 x 2 2 x 1 3 x 1

845

60

Properties of Matrix Multiplication

Matrix Multiplication is not commutative, i.e. AB ≠ BA

Matrix Multiplication is associative, i.e.A(BC) = (AB)C

Matrix Multiplication is distributive, i.e.A(B+C) = AB+AC

Special Types of Matrices

Idempotent MatrixNilpotent MatrixInvolutory Matrix

Transpose of Matrix

Let A be any matrix. The matrix obtained by interchanging rows and columns of A is called the transpose of A and is denoted by A’ or AT.

Properties of Transpose of Matrices

1. The transpose of transposed matrix is equal to the matrix itself, i.e. (A’)’ = A.

2. The transpose of the sum of the two matrices is equal to the transpose of the matrices, i.e. (A+B)’ = A’+B’.

3. The transpose of the product of two matrices is equal to the product of their transposes in the reverse order, i.e.

(AB)’ = B’A’.

Matrix Determinants

A Determinant is a real number associated with a matrix. Only SQUARE matrices have a determinant.

The symbol for a determinant can be the phrase “det” in front of a matrix variable, det(A); or vertical bars arounda matrix, |A| or .3 1

2 4

Determinant of a 2x2 matrix

1 3

-½ 0

1 0 13 2 32

3

Determinant of a 3x3 matrix

-3 8 ¼

2 0 -¾

4 180 11

Imagine crossing out the first row.And the first column.

Now take the double-crossed element. . .And multiply it by the determinant of the remaining 2x2 matrix

33 0 11 1804

3 33 0 11 180 8 2 11 44 4 33 0 11 1804

Determinant of a 3x3 matrix

-3 8 ¼

2 0 -¾

4 180 11

Now keep the first row crossed.Cross out the second column.

•Now take the negative of the double-crossed element.•And multiply it by the determinant of the remaining 2x2 matrix.•Add it to the previous result.

Determinant of a 3x3 matrix

3 33 0 11 180 8 2 11 44 4

Finally, cross out first row and last column.

•Now take the double-crossed element.•Multiply it by the determinant of the remaining 2x2 matrix.•Then add it to the previous piece.

1 2 180 0 44

-3 8 ¼

2 0 -¾

4 180 11695

Computation

Method of Cofactors Also known as the expansion of minors

Method of Minors

Determinant of a 2 x 2 matrix is difference in products of diagonal elements.

711242114

AA

General form for 2 x 2 matrix

dcba

A Then,

A ad - bc

What about larger matrices?

Use method of cofactorsNeed to define a new term, “minor”– Minor of an element aij is the determinant of

the matrix formed by deleting the ith row and jth column

Example

332331

232221

131211

413122321

aaaaaaaaa

A

Minor of a12 = 2 is determinant of the 2 x 2 matrix obtained by deleting the 1st row and 2nd column

332331

232221

131211

413122321

aaaaaaaaa

A

Minor of a12 is

5384312

Minor of a13 = 3 is

4621322

Cofactors

Definition– The cofactor of aij = (-1)i+j x minor

Evaluate cofactors for first three elements of the 3 x 3 matrix

A11(-1)1+1 = 1A12(-1)1+2 =-1A13(-1)1+3 =1

Pattern of signs

+ - +- + -+ - +

Matrix of Cofactors

333231

232221

131211

ccccccccc

C

Determinant obtained by expanding along any row or column of matrix of

cofactors

Determinant of A given by

131312121111 cacacaA

Determinant of A

Element Minor Cofactor Element x Cofactor

a11 = 1 7 7

a12 = 2 -5 -10

a13 = 3 -4 -12

74112

54312

41322

Determinant of A = -15

Determinants of 4 x 4 matrices

Computational energy increases as order of matrix increasesUse pivotal condensation (computer algorithm)

Key Properties of Determinant1. Determinant of matrix and its transpose are

equal.2. If any two adjacent rows(columns) of a

determinant are interchanged, the value of the determinant changes only in sign.

3. If any two rows or two columns of a determinant are identical or are multiple of each other, then the value of the determinant is zero.

4. If all the elements of any row or column of a determinant are zero, then the value of the determinant is zero.

5. If all the elements of any row (or column) of a

determinant are multiplied by a quantity (K), the value of the determinant is multiplied by the same quantity.

6. If each element of a row (or column) of a determinant is sum of two elements, the determinant can be expressed as the sum of two determinants of the same order.

7. The addition of a constant multiple of one row (or column) to another row (or column) leave the determinant unchanged.

8. The determinant of the product of two matrices of the same order is equal to the product of individual determinants.

Adjoint of a Matrix

If A is any square matrix, then the adjoint of A is defined as the transpose of the matrix obtained by replacing the element of A by their corresponding co-factors.Adj.A = Transpose of the cofactor matrix

Inverse Matrix

Inverse of square matrix A is a matrix A-1 that satisfies the following equation– AA-1 = A-1A = I

Steps to success in Matrix Inversion

If the determinant = 0, the inverse does not exist if the matrix is singular.Replace each element of matrix A, by it’s minorCreate the matrix of cofactorsTranspose the matrix of cofactors– Forms the adjoint

Divide each element of the adjoint by the determinant of A.

Matrix InversionPre multiplying both sides of the last equation by A-1, and using the result that A-1A=I, we can get

This is one way to invert matrix A!!!

1 1' 1,C A or A adj AA A

Matrix Inversion

Example

11 12

21 22

131

2 2

3 22 0 inverse exists

1 0

0 12 3

0 2'

1 3

0 10 21 11 32

A A

C CC

C C

C adjA

A adj AA

Properties of Inverse Matrices

If A and B are non-singular matrices of the same order, then (AB)-1 = B-1.A-1

The inverse of the transpose of a matrix is equal to the transpose of the inverse of that matrix, i.e. (A’)-1 = (A-1)’The inverse of the inverse of a matrix is the matrix itself i.e. (A-1)-1 = A

Cramer’s Rule

Given an equation system Ax=d where A is n x n.

1 1 ( ) method of inversex A d adj A dA

1 11 Cramer's Rulex AA

|A1| is a new determinant were we replace the first column of |A| by the column vector d but keep all the other columns intact

Cramer’s Rule

11

n

i ii

d C

1 11x AA

The expansion of the |A1| by its first column (the d column) will yield the expression

because the elements di now take the place of elements aij.

Cramer’s Rule

In general,

11 12 1 1

21 22 2 2

1 2

1n

j nj

n n n nn

a a d aA a a d a

xA A

a a d a

This is the statement of Cramers’Rule

Cramer’s Rule1 2

1 2

5 3 306 2 8x xx x

1 2

11

21

5 3 30 3 5 3028 84 140

6 2 8 2 6 8

84 328

140 528

A A A

Ax

A

Ax

A

Find the solution of

Cramer’s Rule

1 2 3

1 2 3

1 2 3

1 2 3

7 010 2 86 3 2 7

61, 61, 183, 244,

x x xx x xx x x

A A A A

Find the solution of the equation system:

♫ Work this out!!!!

Cramer’s Rule

11

22

23

61 161

183 361

244 461

Ax

A

Ax

A

Ax

A

Solutions:

Note that |A| ≠ 0 is necessary condition for the application of Cramer’s Rule. Cramer’s rule is based upon the concept of the inverse matrix, even though in practice it bypasses the process of matrix inversion.

Rank of a Matrix

The number ‘r’ is called the rank of the matrix A if

1. There exists at atleast one non-zero minor of order r of A

2. Every minor of order (r+1) of A is zero.The rank of a matrix A is denoted by p(A).

Steps to Find Rank of a Matrix1. The given matrix should be a square matrix. If it

is not so, the matrix should be made a square matrix by deleting the extra row or the column.

2. Find the determinant of the square matrix given or obtained after deleting extra row or column.

3. If determinant of the matrix is zero, then take the sub-matrix of the given matrix. Of the determinant of any one of the sub-matrices is not zero, then the order of that sub matrix would be the rank of the given matrix.