2MA101-Linear Algebra

Matrix Theory

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Topics of Unit 1 – Matrix Theory

1 Review of algebra of matrices

2 Rank of matrix

3 Echelon and row reduced echelon form

4 Rank using echelon forms

5 Rank using normal form

6 Inverse by Gauss-Jordan method

7 Solution of system of algebraic simultaneous equations by

Gauss-elimination & Gauss-Jordan method

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Review of Matrix Algebra

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Definition of Matrix

• A matrix is a one-or more dimensional array

• A quantity is usually designated as a matrix by bold face type: A

• The elements of a matrix are shown in square brackets [ ]:

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Row 1 = R1Row 2 = R2Row 3 = R3

Co

lum

n 1

= C1

Co

lum

n 2

= C2

Co

lum

n 3

= C3

Order of a matrix is m rows and n columns = m x nTherefore, given matrix A is of order 3 x 3.

Square Matrices

• Same number as rows as columns.

• If number of rows and columns are not equal then matrix is rectangular matrix.

• Entries mii are called the diagonal entries. The others are called non diagonal entries

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Diagonal Elements

Diagonal Matrix

A diagonal matrix is a square matrix whose non diagonal elements are zero.

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The Identity Matrix

The identity matrix of dimension n, denoted In, is the n x n matrix with 1s on the diagonal and 0s elsewhere.

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Upper & Lower Triangular Matrix

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A square matrix is called lower triangular if all the entries above themain diagonal are zero. Similarly, a square matrix is called uppertriangular if all the entries below the main diagonal are zero.

Vectors as Matrices

• A row vector is a 1 x n matrix.

• A column vector is an n x 1 matrix.

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1 x 3 Row Vector

3 x 1 Column Vector

Transpose of a Matrix

• The transpose of an r x c matrix M is a c x r matrix called MT

• Take every row and rewrite it as a column

• Equivalently, flip about the diagonal

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• Transpose is its own inverse: (MT)T = M for all matrices M

• DT = D for all diagonal matrices D (including the identity matrix I)

Multiplying By a Scalar

• Can multiply a matrix by a scalar.

• Result is a matrix of the same dimension.

• To multiply a matrix by a scalar, multiply each component by the scalar.

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Matrix Multiplication

Multiplying an r x n matrix A by an n x c matrix B gives an r x cresult AB.

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Multiplication: ResultMultiply an r x n matrix A by an n x c matrix B to give an r x c result C = AB.

Then C = [cij], where cij is the dot product of the ith row of A with the jth column of B.

That is:

Chapter 4 Notes 3D MATH PRIMER FOR GRAPHICS & GAME DEV 13

2 x 2 Case

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2 x 2 Example

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Example

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Find C = A × B.

Step 1 : Multiply the elements in the first row of A with the corresponding elements in the first column of B. Add the products to get the element C 11

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Step 2 : Multiply the elements in the first row of A with the corresponding elements in the second column of B. Add the products to get the element C 12

Step 3 : Multiply the elements in the second row of A with the corresponding elements in the first column of B. Add the products to get the element C 21

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Step 4 : Multiply the elements in the second row of A with the corresponding elements in the second column of B. Add the products to get the element C 22

3 x 3 Case

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3 x 3 Example

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Matrix Multiplication Facts

• Not commutative: in general AB BA.

• Associative:

(AB)C = A(BC)

• Associates with scalar multiplication:

k(AB) = (kA)B =A(kB)

• (AB)T = BTAT

• (M1M2M3…Mn)T = Mn

T…M3TM2

TM1T

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Row Vector Times Matrix Multiplication

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Matrix Times Column Vector Multiplication

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Determinate of a Matrix

• The determinate of a square matrix is a scalar quantity that has some uses in matrix algebra. Finding the determinate of 2 × 2 and 3 × 3 matrices can be done relatively easily:

• The determinate is designated as |A| or det(A)

• 2 × 2:

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Determinate of a Matrix

• 3 × 3:

• Similar for larger matrices.

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If A is a square matrix

Cofactor method

The minor, Mij, of entry aij is the determinant of the submatrix

that remains after the ith row and jth column are deleted from A.

The cofactor of entry aij is Cij=(-1)(i+j) Mij

31233321

3331

2321

12 aaaaaa

aaM

3331

2321

1212aa

aa MC

aaa

aaa

aaa

A

333231

232221

131211

What is a cofactor?

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Sign of cofactor

What is a cofactor?

-

--

-

Find the minor and cofactor of a33

4140201

42M 33

Minor

4MM)1(C 3333)33(

33

Cofactor

2

01A

21-

4

3-42

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Inverse of a Matrix

• Some square matrices have an inverse

• If the inverse of a matrix exists (designated by -1 superscript), then

where I is the identity matrix

OR

𝐴−1 =1

𝐴𝐴𝑑𝑗 𝐴

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Inverse of a Matrix

• The inverse of a 2X2 matrix is easy to find:

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Inverse of a Matrix

• Example: find inverse of A:

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Inverse of a Matrix

The inverse of an nn matrix A is an nn matrix B having the property

that

AB = BA = I

B is called the inverse of A and is usually denoted by A-1 .

If a square matrix has an inverse, it is said to be invertible or

nonsingular.

If it doesn’t possess an inverse, it is said to be singular.

Example: The inverse of is because

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11

10

01

10

11

10

11

10

11

10

11

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Next Lecture : System of Linear Equations & Matrix