Mathematics Lecture1

SSE 1523 MATEMATIK UNTUK JURUUKUR

Lecture 1 Matrix Operation Algebraic Properties Types of Matrices

Linear System •

One of the most frequently recurring practical problems in many fields of study such as geomatics is that of solving a system of linear equations.

a1x1 + a2 x2 + ... + an xn = b

•

If the linear system has no solution, it is said to be inconsistent.

•

If it has a solution, it is called consistent. If b1=b2=…=bm=0, then it is called a homogeneous system.

•

To find solutions to a linear system, use the technique called the method of elimination.

•

One method of elimination is Matrices.

Matrices •

•

What is a general meaning of MATRIX – “A concise and useful way of uniquely representing and working with linear transformations.” Notation ... A1m ⎤ ⎡ A11 A12 A13 ... ⎥ ⎢A ... ... A A A 22 23 2m ⎥ ⎢ 21 ⎢ A31 A32 A33 ... ... A3 m ⎥ A=⎢ ⎥ ... ... ... ... ... ... ⎥ ⎢ ⎢ ... ... ... ... An −1, m −1 An −1, m ⎥ ⎥ ⎢ A A A ... A A ⎢⎣ n1 n2 n3 n , m −1 n ,m ⎥ ⎦

Matrix Operations - Scalar Scalar - Matrix Addition Subtraction Multiplication Division

Matrix Operations - Matrix •

Matrix – Matrix – Transpose – Determinant – Matrix Addition – Matrix Multiplication – Matrix Inverse

• A matrix is a rectangular array of objects (usually numbers). • An m×n (“m by n”) matrix has exactly m horizontal rows, and n vertical columns.(sometimes use i row and j column • An n×n matrix is called a square matrix, whose order or rank is n (same number of row and column, e.g. 2x2, 3x3, 4x4 matrices A 3 rows ×2 columns matrix

⎡2 3 ⎤ ⎢5 − 1⎥ ⎢ ⎥ ⎢⎣7 0 ⎥⎦

⎡ A11 ⎢A A = ⎢ 21 ⎢ ... ⎢ ⎣A1n

A12 ... A1n ⎤ A22 ... A2n ⎥⎥ ... ... ... ⎥ ⎥ A2n ... Ann⎦

Examples •

These are valid matrices

⎡1 ⎢0 ⎣

0⎤ 1 ⎥⎦

⎡a ⎢c ⎣

b⎤ e ⎥⎦

⎡1 2 3⎤ ⎢0 1 0⎥ ⎦ ⎣

⎡ 11 ⎤ ⎢ 7 ⎥ ⎢ ⎥ ⎢8 9⎥ ⎢ 10 ⎥ ⎦ ⎣ >> A can be constructs an (n x n) “identity” matrix. 1 0 0 0 1 0 0 0 1 >> A = ones(3,2) (ones(m,n) constructs an m x n array of 1’s) A= 1 1 1 1 1 1 >> A = zeros(2,3)(zeros(m,n) constructs an m x n array of 0’s) A= 0 0 0 0 0 0 •

These have problems ⎡1 2⎤ ⎡ 4 ⎤ ⎢3 ⎥ ⎢5 6⎥ ⎣ ⎦ ⎦ ⎣

Matrix Short notation

A3 X 3

⎡ a11 a12 = ⎢⎢a21 a22 ⎢⎣ a31 a32

Order = 3x3

What are the dimensions of W?

a13 ⎤ a23 ⎥⎥ a33 ⎥⎦

This is the 2,3 element, or element23

3 x 4 or 4 x 3 …

2 ⎞ ⎛1 0 ⎜ ⎟ ⎜2 −3 4 ⎟ W =⎜ 3 33 5 ⎟ ⎜ ⎟ ⎜ 4 2 − 2⎟ ⎝ ⎠

What is w32 + w21 ? Is it a) 4 b) 5 c) 6 w21 = 2

w32 = 3

A matrix with m rows and n columns is said to have SIZE m × n.. A matrix with only one column is a

⎡ a11 a12 ⎤ A=⎢ = aij ⎥ ⎣a21 a22 ⎦

[ ]

COLUMN MATRIX, and a matrix with only one row is a ROW MATRIX. Two matrices are EQUAL if they have the same size and their corresponding elements are equal

Applications of Matrices Many of applications, including: • Solving systems of linear equations • Computer Graphics, Image Processing • Models within many areas of Computational Science & Engineering • Quantum Mechanics, Quantum Computing • Survey adjustment (network), etc

⎛ b1 ⎜ ⎜ b2 b = ⎜ b3 ⎜ ⎜ ... ⎜b ⎝ n

cT = (c1 c2

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

c3 ... cn )

ROW AND COLUMN VECTORS IN MATRICES Matrix Am x n = (aij)m x n = Column vector j =

•

a1j a2j … amj

Scalar matrix: diagonal matrix with same non-zero values on entire main diagonal

Zero matrix

⎡0 ⎢0 Z =⎢ ⎢M ⎢ ⎣0

0 L 0⎤ 0 L 0⎥⎥ M O M⎥ ⎥ 0 L 0⎦

a11 a21 … am

a12 a22 … am

1

2

… … … …

Row vector i =

⎡5 0 0 ⎤ ⎢0 5 0 ⎥ ⎢ ⎥ ⎢⎣0 0 5⎥⎦

a1n a2n … am n

ai1 ai2 … ain

Diagonal Matrix Diagonal matrix: one or more non-zero values on main diagonal (topleft to bottom-right); all zeros on off-diagonal

⎡ λ1 ⎢0 ⎢ λ = ⎢ ... ⎢ ⎢0 ⎢⎣ 0

⎡1 ⎢0 ⎢ I = ⎢ ... „Unity or identity matrix is ⎢ „special case of scalar matrix ⎢0 „where diagonal values = 1 ⎢⎣ 0

Identity Matrix

•

0

λ2

... ...

0 0

...

...

...

0

...

λ n −1

0

...

0

0

...

0

1 ... 0 0

... ... ... ...

0 ... 1 0

Equality: two matrices of same order and with all equal elements

⎡1 2⎤ A=⎢ ⎥ 3 2 ⎣ ⎦

and

⎡1 B=⎢ ⎣y

–

Assume

–

Then A = B implies that x=2 and y=3

x⎤ 2 ⎥⎦

0⎤ 0 ⎥⎥ ... ⎥ ⎥ 0⎥ λ n ⎥⎦ 0⎤ 0 ⎥⎥ ... ⎥ ⎥ 0⎥ 1 ⎥⎦

⎡1 ⎢0 ⎢ ⎢⎣ 0

0 5 0

0⎤ 0 ⎥⎥ 8 ⎥⎦

⎡ L11 ⎢L Triangular Matrices - I ⎢ 21 Lower Triangular Matrix ⎢ L31 L=⎢ ⎢ ... ⎢ ... ⎢ ⎢⎣ Ln1 Triangular Matrices - II

0 L22

0 0

... ...

... ...

... ...

L32 ...

L33 0 ... ...

... ...

... ...

... Ln 2

... ...

... Ln −1,n − 2 ... ...

Ln −1,n −1 Ln ,n −1

Upper Triangular Matrix

⎡U11 U12 U13 ... ⎢ 0 U U 23 ... 22 ⎢ ⎢ 0 0 U 33 U 34 U=⎢ ... ... ... ⎢ ... ⎢ ... ... ... ... ⎢ ... ... ... ⎢⎣ 0

... ...

... ...

... ...

... ...

0 U n −1,n −1 ...

0

U1n ⎤ U 2 n ⎥⎥ U 3n ⎥ ⎥ ... ⎥ U n −1,n ⎥ ⎥ U n ,n ⎥⎦

0 ⎤ 0 ⎥⎥ 0 ⎥ ⎥ ... ⎥ 0 ⎥ ⎥ Ln ,n ⎥⎦

Symmetric matrix

Aij = A ji

Anti-symmetric matrix

Aij = − A ji

Positive definite matrix

xT Ax > 0

Positive semi-definite matrix

xT Ax ≥ 0

Matrix transpose

AijT = A ji

The Inverse

A −1A = AA −1 = I where A −1 is called the inverse of A One often encounters formulae such as the following:

y = B −1 (I + 2 A )b z = B − 1 ( I + 2 A )( C − 1 + A ) b + d Simultaneous Row and Column Matrix Multiplication ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣

× → × → × → × → × →

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

⎡ ⎤ ⎢ ⎥ × ⎢ ⎥ ⎡ ⎢× × × × ×⎥ ⎢× ⎢ ⎥ ⎢ ⎢↓ ↓ ↓ ↓ ↓⎥ = ⎢× ⎢ ⎥ ⎢× ⎢ ⎥ ⎢ ⎢ ⎥ ⎢⎣× ⎢ ⎥ ⎣ ⎦

× × × ×⎤ × × × ×⎥⎥ × × × ×⎥ ⎥ × × × ×⎥ × × × ×⎥⎦

Addition 1

2

3

4

A

+

+

5

6

7

8

B

=

=

6

8

10 12

C

Addition Conformability To add two matrices A and B: • # of rows in A = # of rows in B • # of columns in A = # of columns in B •

each nm elements added together to produce a new matrix of same order – Addition not defined for matrices of different orders

The SUM of two matrices of the same size, m × n, is an m × n matrix whose elements are the sum of the corresponding elements of the two given matrices. Addition is not defined for matrices with different sizes. Matrix addition is commutative, if: A + B = B + A, and associative, if: (A + B) + C = A + (B + C). ADDITIVE IDENTITY: A + 0 = 0 + A = A ADDITIVE INVERSE: A + (-A) = (-A) + A = 0 A matrix with all elements equal to zero is called a ZERO MATRIX.

Addition: (aij) m × n + (bij) m × n = (aij+bij) m × n

(2 i + 3 j) + (4 i + 5 j)

[2 3] +[4 5] = [6 8]

⎡a11 +b11 a12 +b12 ⎤ A+ B = ⎢ = aij +bij ⎥ ⎣a21 +b21 a22 +b22⎦

[

]

Exercise

⎛3 5 1/2 A=⎜⎜ ⎝4 −1 2

4⎞ 0⎠

⎛2 ⎜ A = ⎜3 ⎜5 ⎝

⎛2 ⎜ A = ⎜3 ⎜5 ⎝

⎛−1 4 0 3⎞ B=⎜⎜ ⎝ 0 2 1 5⎠ 0 ⎞ ⎛0 ⎜ ⎟ − 6 ⎟B = ⎜ 6 ⎜3 1 ⎟⎠ ⎝

0 ⎞ ⎛0 ⎟ ⎜ − 6 ⎟B = ⎜ 3 ⎜ 1 ⎟⎠ ⎝

− 6⎞ ⎟ 7 ⎟ 0 ⎟⎠

− 3⎞ ⎟ 1 ⎟ ⎟ ⎠

From the above matrix elements, find the following tasks: A + B, B + A, 3A, 2B, (3A + 2B), (2B + 3A)

COMBINED PROPERTIES •

LEFT DISTRIBUTIVE:

A(B + C) = AB + AC

•

RIGHT DISTRIBUTIVE:

(B + C)A = BA + CA

EQUALITY •

ADDITION:

If A = B then A + C = B + C.

•

LEFT MULTIPLICATION:

If A = B, then CA = CB.

•

RIGHT MULTIPLICATION:

If A = B, then AC = BC.

Subtraction of matrices • Example

⎡4−(−1) −3−2 1−3 ⎤ ⎢ 0−6 5−(−7) −2−9⎥ ⎢ ⎥ ⎢⎣ 5−0 −6−(−4) 0−8 ⎥⎦

A-B=C

⎡4 −3 1 ⎤ ⎡−1 2 3⎤ ⎢0 5 −2⎥ −⎢ 6 −7 9⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣5 −6 0 ⎥⎦ ⎢⎣ 0 −4 8⎥⎦

=

Subtraction Conformability • • •

⎡ 5 ⎢ −6 ⎢ ⎢⎣ 5

−5 12 −2

−2 ⎤ ⎥ − 11⎥ − 8 ⎥⎦

To subtract two matrices A and B: # of rows in A = # of rows in B # of columns in A = # of columns in B each nm elements subtracted with each other to produce a new matrix of same order Addition not defined for matrices of different orders

Alternate definition of subtraction of matrices: „

The definition of subtract of two real numbers a and b is : a – b = a + (-1)b or a plus the opposite of b.

„

If A and B are two matrices of the same dimensions, then: A – B = A + (-1)B, where -1 is a scalar.

⎡1 2 ⎤ ⎡ 5 6⎤ ⎡1 2⎤ ⎢3 4 ⎥ − ⎢ 7 8 ⎥ = ⎢3 4 ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎡ 5 6 ⎤ ⎡1 2 ⎤ ⎡ −5 −6⎤ +(−1) ⎢ =⎢ +⎢ ⎥ ⎥ ⎥ 7 8 3 4 7 8 − − ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎡ −4 −4 ⎤ ⎢ −4 −4 ⎥ ⎣ ⎦

Exercise

⎛3 5 1/ 2 A=⎜⎜ ⎝4 −1 2

⎛−1 4 0 3⎞ ⎟⎟ B = ⎜⎜ ⎝ 0 2 1 5⎠

4⎞ ⎟⎟ 0⎠

⎛2 ⎜ A = ⎜3 ⎜5 ⎝

⎛2 ⎜ A = ⎜3 ⎜5 ⎝

0 − 6 1

⎞ ⎛0 ⎟ ⎜ ⎟B = ⎜6 ⎟ ⎜3 ⎠ ⎝

0 ⎞ ⎛0 ⎟ ⎜ − 6 ⎟B = ⎜ 3 ⎜ 1 ⎟⎠ ⎝

− 6 7 0

⎞ ⎟ ⎟ ⎟ ⎠

− 3⎞ ⎟ 1 ⎟ ⎟ ⎠

From the above matrix elements, find the following tasks: A - B, B - A, (3A - 2B), (2B - 3A)

Adding/subtracting a scalar k (a constant): A = W + k = k+W means W with k added to each element ie aij = wij + k

eg

⎛ 1 2 3⎞ ⎛ 2 3 4⎞ ⎟⎟ + 1 = ⎜⎜ ⎟⎟ A = ⎜⎜ ⎝ 4 5 6⎠ ⎝ 5 6 7⎠

Matrix plus/minus a matrix: both must be of the same size and we add each element together (point-by-point or point-wise operation) ie if C = A + B then cij = aij + bij

⎛ 1 2 3⎞ ⎛ 1 1 1 ⎞ ⎛ 1+1 2 +1 3 +1 ⎞ ⎛ 2 3 4⎞ eg ⎜ ⎜ 4 5 6 ⎟⎟ + ⎜⎜ 2 2 2 ⎟⎟ = ⎜⎜ 4 + 2 5 + 2 6 + 2 ⎟⎟ = ⎜⎜ 6 7 8 ⎟⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Multiplication of a Row Matrix and a Column Matrix • In general procedure of matrix multiplication, the concept of the product of a row matrix by a column matrix is important. • A row matrix consists of a single row of numbers while a column matrix consists of a single column of numbers. If the number of columns of a row matrix equals the number of rows of a column matrix, the product of a row matrix and column matrix is defined. • Otherwise, the product is not defined. For example, a row matrix consists of 1 row of 4 numbers so this matrix has four columns. It has dimensions • 1 X 4. This matrix can be multiplied by a column matrix consisting of 4 numbers in a single column (this matrix has dimensions 4X1.

Matrix Product (Multiplication) •

If A is an m x p matrix and B is a p x n matrix, the matrix product of A and B denoted by AB is an m x n matrix whose element in the ith row and jth column is the real number obtained from the product of the ith row of A and the jth column of B. If the number of columns of A does not equal the number of rows of B, the matrix product AB is not defined. note:

number of rows in A must equal the number of columns in B

note:

A•B≠B•A (aij) m × n . α = (aij . α)m × k (α is a constant) (aij) m × n . (bij) n × k = (cij) m × k where cij = ai1 . b1j + ai2 . b2j + .. + ain.bnj

„Matrix

multiplication is not commutative AB ≠ BA

„Matrix

multiplication is associative A(BC) = (AB)C

„For

the matrix multiplication the zero factor property does not hold.

Basic Operations •

Vector multiplication or Inner products of a row vector * column vector (always of same orders): r1*c1 + r2*c2+ . . . rn*cn

⎡d ⎤ [a b c]⎢⎢ e ⎥⎥ = (a * d ) + (b * e) + (c * f ) = k ⎢⎣ f ⎥⎦ •

Matrix*matrix multiplication: – Inner product of First row (first matrix) * first column (second matrix) → a1 – First row * second column → a2 – Second row * first column → b1 – Second row * second column → b2

⎡a b ⎤ ⎡ g ⎢c d ⎥ x⎢h ⎣ ⎦ ⎣

i ⎤ ⎡ag + bh ai + bj ⎤ =⎢ ⎥ j ⎦ ⎣ cg + dh ci + cj ⎥⎦

Matrix-Constant Multiplication if A=Bk = kB, where k is a constant: multiply all elements in B by k ie aij= kbij

⎛ 1 2 3 ⎞ ⎛ 1× 3 2 × 3 3 × 3 ⎞ ⎛ 3 6 9 ⎞ ⎟⎟ ⎟⎟3 = ⎜⎜ ⎟⎟ = ⎜⎜ A = ⎜⎜ ⎝ 4 5 6 ⎠ ⎝ 4 × 3 3 × 3 6 × 3 ⎠ ⎝12 15 18 ⎠ Multiplication Conformability • • •

Regular Multiplication

5

6

7

8

2

3

4

=

23

34

31

46

To multiply two matrices A and B: # of columns in A = # of rows in B

A mxn

•

x

1

Multiply: A (m x n) by B (n by p)

x

B nxp

=

C mxp

MULTIPLICATIVE IDENTITY: AI = IA = A MULTIPLICATIVE INVERSE: If A is a square matrix and A-1 exists, then AA-1 = A-1A = I.

Get A * B B A ⎡ 3 1 −1⎤ ⎢2 0 3 ⎥ ⎣ ⎦

⎡1 ⎢3 ⎢ ⎢⎣ − 2

6⎤ − 5 ⎥⎥ 4 ⎥⎦

8 9 ⎡ ⎤ AB = ⎢−4 24⎥ ⎣ ⎦

Since A is a 2 x 3 matrix and B is a 3 x 2 matrix, the solution of product AB will be a 2 x 2 matrix. 1. Multiply first row of A by first column of B: 3(1) + 1(3) +(-1)(-2)=8 2. First row of A times second column of B: 3(6)+1(-5)+ (-1)(4)= 9 Get B * A . Give some comments

Matrix-Matrix Multiplication if A=BC, to get the value in the i’th row and j’th column in A, take the i’th row in B, multiply each element by the corresponding element in the j’th column in C, and add them together

⎛ 1 1 2⎞ ⎟ ⎛ (1×1) + (2 × 2) + (3× 3) 1+ 2 + 3 2 + 4 + 6 ⎞ ⎛ 1 2 3⎞⎜ ⎟⎟ ⎟⎟⎜ 2 1 2⎟ = ⎜⎜ ⎜⎜ ⎝ 4 5 6⎠⎜ 3 1 2⎟ ⎝ (4 ×1) + (5× 2) + (6 × 3) 4 + 5 + 6 8 +10 +12⎠ ⎠ ⎝ n columns ⎛ 14 6 12 ⎞ m rows ⎟⎟ = ⎜⎜ n rows ⎝ 32 15 30 ⎠ p columns

so inner dimensions must match ie n

(m x n) (n x p) Æ m x p

Formula:

aij = ∑ bik ckj k =1

Solution using matrix multiplication • We represent the number of each model sold using a row matrix (4X1) and we use a 1X4 column matrix to represent the sales price of each model. When a 4X1 column matrix is multiplied by a 1X4 row matrix, the result is a 1X1 matrix of a single number. Example: ⎡12,500⎤ ⎢11,800⎥ ⎥ = [10(12,500) +5(11,800) +8(15,900) + 3(25,300)] = [387,100] [10 5 8 3] ⎢⎢ 15,900⎥ ⎢ ⎥ ⎣25,300⎦

Solution using matrix multiplication EXAMPLE: the product of a 2 x 4 matrix with a 4 x 3 will produce result 2 x3 matrix . First, the number of columns of the matrix on the left equals the number of rows of the matrix on the right so matrix multiplication is defined. A row by column multiplication is performed three times to obtain the first row of the product: 70 80 90.

(1x1+2x4+3x7+4x10): 1x1+2x5+3x8+4x11:1x3+2x6+3x9+4x12 Undefined matrix multiplication

• Why is this matrix multiplication not defined? The answer is that the left matrix has three columns but the matrix on the right has only two rows. To multiply the second row [4 5 6] by the third column, 3 there is no number to pair with 6 to multiply, i.e. 4 x 3, 5 x 7, 6 x ?

• Matrix*matrix multiplication: – Multiplication only defined (conformable) if # columns in left factor = # rows in right factor – Multiplication not commutative (even subset of cases in which conformable in both directions) – Multiplication only defined (conformable) if # columns in left factor = # rows in right factor

⎡1 2⎤ ⎡2⎤ ⎡ (1x2) + (2x3) ⎤ ⎡ 8 ⎤ ⎢3 4⎥ x⎢3⎥ = ⎢(3x2) + (4x3)⎥ = ⎢18⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎡2⎤ ⎡1 2⎤ BUT ⎢ ⎥ x⎢ is not defined. ⎥ ⎣3⎦ ⎣3 4⎦

– Multiplication not commutative (even subset of cases in which conformable in both directions) ⎡2 ⎢3 ⎣ ⎡2 ⎢0 ⎣

⎡1 ⎢2 ⎢ ⎢⎣3

0⎤ ⎡2 1⎤ ⎡(2 x 2) + (0 x0) =⎢ x⎢ ⎥ ⎥ 2⎦ ⎣0 5⎦ ⎣ (3 x 2) + (2 x0) 1⎤ ⎡2 0⎤ ⎡ (2 x 2) + (1x3) x⎢ =⎢ ⎥ ⎥ 5⎦ ⎣3 2⎦ ⎣(0 x 2) + (5 x3)

(2 x1) + (0 x5)⎤ ⎡4 2 ⎤ =⎢ ⎥ (3 x1) + (2 x5) ⎦ ⎣6 13⎥⎦ (2 x0) + (1x 2) ⎤ ⎡ 7 2 ⎤ =⎢ ⎥ (0 x0) + (5 x 2)⎦ ⎣15 10⎥⎦

2⎤ ⎡(1x2)+(2x1) (1x1)+(2x2) (1x0)+(2x2)⎤ ⎡4 5 4⎤ ⎡2 1 0⎤ ⎢ ⎥ ⎥ ⎢ ⎥ = (2x2)+(1x1) (2x1)+(1x2) (2x0)+(1x2) = 5 4 2 1 x⎢ ⎥ ⎥ ⎣1 2 2⎦ ⎢ ⎥ ⎢ ⎥ 2⎥⎦ ⎢⎣(3x2)+(2x1) (3x1)+(2x2) (3x0)+(2x2)⎥⎦ ⎢⎣8 7 4⎥⎦

• Recall scalar matrix; now we see why it is interesting:

⎡2 ⎢0 ⎣

0⎤ ⎡2 x⎢ ⎥ 2⎦ ⎣0

1⎤ ⎡(2x2)+(0x0) (2x1)+(0x5)⎤ ⎡4 2⎤ =⎢ =⎢ ⎥ ⎥ ⎥ 5⎦ ⎣(0x2)+(2x0) (0x1)+(2x5)⎦ ⎣0 10⎦

– Multiplying by scalar matrix same as multiplying by a scalar •

Recall identity matrix; now we see why it is interesting:

⎡1 ⎢0 ⎣

0⎤ ⎡2 x⎢ ⎥ 1⎦ ⎣0

1⎤ ⎡(1x2)+(0x0) (1x1)+(0x5)⎤ ⎡2 =⎢ =⎢ ⎥ ⎥ 5⎦ ⎣(0x2)+(1x0) (0x1)+(1x5)⎦ ⎣0

– Multiplying by identity matrix same as multiplying by 1 in regular algebra

1⎤ ⎥ 5⎦

Get the following solutions of : AB, BA. 2A +2B, Give comment on the results

⎛1 ⎜ A = ⎜3 ⎜5 ⎝

⎛p A = ⎜⎜ ⎝r

⎞ ⎟ ⎛7⎞ ⎟ B = ⎜⎜ ⎟⎟ 8⎠ ⎝ ⎟ 6⎠ 2 4

q⎞ ⎛ s ⎟⎟ B = ⎜⎜ s⎠ ⎝− r

− q⎞ ⎟⎟ p ⎠

Get the solution of IA and AI

⎛ p q⎞ ⎛1 0⎞ ⎟⎟ ⎟⎟ A = ⎜⎜ I = ⎜⎜ ⎝ r s⎠ ⎝0 1⎠

CLASS EXERCISE – GIVE SOME COMMENTS

Transpose of a matrix • If the rows and columns of a matrix are interchanged, i.e. – The first row becomes the first column, – The second row becomes the second column, – The third row becomes the third column etc,

• The new matrix so formed is called the transpose of the original matrix. • If A is the original matrix, its transpose is denoted by A or AT.

⎡ 4 6⎤ If A = ⎢⎢7 9⎥⎥ ⎢⎣2 5⎥⎦

Then AT:

⎡ 4 ⎢ 6 ⎣

7 9

2 ⎤ ⎥ 5 ⎦

Exercise: Given::

⎡ 2 7 6⎤ A=⎢ ⎥ ⎣ 3 1 5⎦

⎡4 0⎤ B = ⎢⎢3 7 ⎥⎥ ⎢⎣1 5⎥⎦

Then calculate A*B and (A*B)T: