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Is a set of equations containing two or moreunknowns having similar solution set.
Linear SystemsIs a set oflinear equations containing two or
more unknowns having similar solution set.
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Examples:1. 2.
3. 4.
943 yx
1087 zyx
0852 yx
254 zxx
3825 xyx
0348 yx
626 zyx
7 yx
03742 zyx
10 zyx
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1. Independent or consistent system
System of equations in two or more variables represented by linesintersecting at a common point.
Have finite number of solutions represented by the points of
intersection of the lines.2. Inconsistent system
System of equations in two or more variables represented by non-intersecting curves.
System with no solution3. Dependent system
System of equations in two or more variables represented by curvescoincident with one another.
System with infinite number of solutions.
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1. 2.
3. 4.
5.
8yx
15zy3x4
1yx2
2z2yx 422 zyx
62 yx
9y6x8
1)zy(3
1x
5x3y
1)2(2
1 xzy
10y2x6
2)yx2(4
1z
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Graphical Method of Solving a System of Linear EquationsThe graphical method of solving a system of linear equations is a methodthat determines the solutionin terms of the common point(s)or thepoint(s) of intersectionamong the graphs representing each of theequations in the system.
The following are the basic steps to be followed:
1. Draw the graphs associated to the equations of the system.
2. Determine the common point or the point of intersection among the graphs.3. Read the coordinates of the point giving the solution ( x , y ).
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Find the solution to each of the given system by the graphical method.
1. 2.
1. S (3, 1) 2. S (1, 2)
9y3x2 13y4x3
X Y1 Y20 3 13/4
1 7/3 5/2
2 5/3 7/4
3 1 1
EQ1
EQ26
11
2
y
3
x
4
3
3
y
4
x
EQ1EQ2
X Y1 Y2
0 11/3 9/4
1 3 3
2 7/3 15/4
3 5/3 9/2
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Analytical MethodsElimination Of a Variable by Addition/Subtraction
This is an analytical method of solving a system ofequations that eliminates a variable addition/subtraction of
multiple equations.
Elimination Of a Variable by Substitution
This is an analytical method of solving a system ofequations that eliminates a variable by replacing one of thevariables in one of the equations by an equal expressionsobtained from the other equation.
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At the end of the lesson, the student is
expected to be able to:
Define matrix
Identify different types of matrices.
Perform operations on matrices.Define determinant of matrix.
Evaluate determinant of a square matrix.
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Matrix is a rectangular array of elements
arranged inm rows andn columns, and is
enclosed by a pair of parenthesis ( ), braces [ ]
or brackets { }. The elements maybe
numbers, variables.
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Matrix can be written in the general form:
mna...m2am1a
............2n
a...22
a21
a1na...12a11a
)ij
(aA
row column
Uppercase letterLower case letter Listed elements
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2. Row Vector or Row Matrix
6402A
is a matrix with only one row andn columns
(1xn).
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3. Square Matrix
422
303
012
A
is a matrix with equal number ofrows and
columns.
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4. Symmetric Matrix
423
201
312
A
is a matrix withaij
= aji
.
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5. Diagonal Matrix
400
070
002
A
is a square matrix with elements above and
below the main diagonal is zero.
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6. Identity Matrix or Unit Matrix
100
010
001
A
is a diagonal matrix with all elements on the
main diagonal is equal to 1.
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7. Upper Triangular Matrix
800
320
236
A
is a matrix with all elements below the main
diagonal is equal to 0.
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8. Lower Triangular Matrix
834
027
006
A
is a matrix with all elements above the main
diagonal is equal to 0.
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9. Zero One Matrix
011
00
11
0
0
A
is a matrix consisting ofzeros and ones only
as entries.
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10. Transpose of a Matrix
011
0011
0
0
A
The transpose of a matrix , denoted by AT , is
obtained by interchanging the rows and
columns of the matrix.
000
101101TA
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A. Matrix Equality
ijbaij
BA
A matrix of the same dimension (mxn) if all
elements of the first matrix is equal to its
respective element on the second matrix
for all is and js
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Find the values of a, b, c, and d so thateach of the given statements would be true:a. b.
01
2
031
522
c
ba
2
7
2
1
24
47
20
01
b
d
a
c
=
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B. Matrix Addition
ijbac ijij
BAC
The sum of matrices of the same dimension
(mxn) is the sum of all elements of the same
position. for all is and js
44
92
0422
4531
02
43
42
51
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C. Scalar Multiplication
84
102
42
512
)( ijakAk
The scalar product of matrix is obtained by
multiplying a constant k to every elemnt of
the matrix
for all is and js
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Properties of Matrix Addition and Scalar Multiplication
If A, B, C and O (zero matrix) are m x n matrices and c and d are scalar
numbers, then the following hold true.
1. A + B = B + A Commutative Property of Addition
2. A + (B + C) = (A + B) + C Associative Property of Addition
3. cdA = c(dA) Associative Property of Scalar
Multiplication
4. IA = A Scalar Identity
5. c(A + B) = cA + cB Distributive Property6. (c +d) A = cA + dA Distributive Property
7. A + O = A Identity Property of Addition
Note: The difference AB = A + (-B)
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C. Matrix Multiplication or Vector Product
)()()( kxnxmxkmxn
AxBC
A vector product of matrices A and B can be
obtained if the number ofcolumns of the
multiplicand is equal to the number ofrows
of the multiplier
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C. Matrix Multiplication or Vector Product
p
kkjikij bac 1
7120
5712
1127
130
211
307
013
322
401
x
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C. Matrix Multiplication or Vector Product
p
kkjikij bac 1
7120
5712
1127
130
211
307
013
322
401
x
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C. Matrix Multiplication or Vector Product
p
kkjikij bac 1
7120
5712
1127
130
211
307
013
322
401
x
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Let A, B, and C be matrices with sizes so thatthe given expressions are all defined, and let c
be a real number.
|1. Multiplication is not commutative.
2. A(BC) = (AB) C
3. c(AB) = (cA)B = A (cB)
4. A(B+C) = AB + AC5. (B+C) A = BA + CA
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Suppose A, B, are mxn matrices, C is an nxpmatrix, c is a real number.
|1. (A+B)T = AT + BT.
2. (AC)T = CT AT
3. (AT) T = A
4. (cA)T = cAT
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Given the following matrices, perform
the indicated operations
a. A *B -2A b. B*AT + 2( B)
97
83
52
31BA
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)det(AA
A scalar quantity associated with a square
matrix.
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12
53A
The determinant of an order 2 matrix (2 x 2 matrix) can be
obtained by taking the difference of the products of the
diagonal elements.
40
21B
|A| = det(A) = 3*1 -2*5
|A| = -7|B| = det(B) = -1*4 -0*2
|A| = -4
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12
03B
Evaluate the following determinants
1. 4.
2. 5.
3. 6.
03
12A
41
32C
18
24D
21
53E
12
510F
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For a 3x3 matrix, the determinant is:
333231
232221
131211
||aaaaaa
aaa
A
)()()(|| 312232121333213123123223332211 aaaaaaaaaaaaaaaA
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The determinant of an order 3 matrix (3 x 3 matrix) can beobtained by doing the following steps:
1. Copy the first two columns of the given matrix as the
4th and 5th columns.
2. Multiply the downward diagonal elements of theresulting matrix. Find the sum of these three products.
3. Multiply the upward diagonal elements of the matrix.
Find the sum of these three products.
4. Subtract the result of (2) by the result of (3). This is the
determinant.
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221
103
212
A
Evaluate the following determinant:
21
03
12
|A| = det(A) = (2*0*-2+-1*1*1+2*3*2)
(1*0*2+2*1*2+-2*3*-1)
|A| = 1
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The minor of a matrixmij about an elementaij is asubmatrix of A where the ith row and jth column has
been removed.
The cofactor of a Matrix about the elementaij is the
determinant of the minormij preceded by
(negative) if(i+j) is odd.
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For any square matrix of order n>=3, the determinant can beevaluated using:
a) Co-factor Expansion about a column:
for all i, for a given j
b) Co-factor Expansion about a row:
for all j, for a given i ,
where: aij is the element in the ith-jth position
Aij is the cofactor of aij.
m
i
ijijAaA1
||
n
j
ijijAaA1
||
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221
103
212
A
Evaluate the following determinant:
Column aij mij Sign Aij aij*Aij
1 3 - -(-2) 6
2 0 + +(-6) 0
3 1 - -(5) -5
TOTAL 1
Selecting row 2
21
22
22
21
21
12
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110
322
111
B
Evaluate the following determinants using both methods
1. 3.
2. 4.
204
121
211
A
111
422
301
D
414123
031
F