SETASIGNThis e-book
Tutorial 2.13 Find the product of given matrices
Two matrices (of any dimension R × C) may be multiplied if the number of columns (which run up and down) in the first matrix equals the number of rows (which run left to right) in the second matrix.
The two matrices presented below may be multiplied together. The order (R × C) of the first matrix is 3 × 2 while the order (R × C) of the second matrix is 2 × 4. Since the number of columns in the first matrix is equal to the number of rows in the second matrix, the given matrices may be multiplied as presented.
a b s t u v c d w x y z e f
×
However, if these same matrices are reversed, the order (R × C) of the first matrix is 2 × 4 while the order (R × C) of the second matrix is 3 × 2. Since the number of columns in the first matrix no longer equals the number of rows in the second matrix, the given matrices cannot be multiplied as presented.
s t u v a b w x y z c d
e f
×
Therefore, it is very important that we pay attention to the dimensions of the given matrices and in the order in which we pose a matrix multiplication problem.
When we multiply two matrices that are appropriately presented, we take the elements (members, entries) from the first row of the first matrix, multiply them with the respective elements (members, entries) from each of the columns from the second matrix and then add the products together. Notice how the product of the matrices from above is achieved:
a b as bw at bx au by av bz s t u v
c d cs dw ct dx cu dy cv dz w x y z
e f es fw et fx eu fy ev fz
+ + + +
× = + + + +
+ + + +
NOTE: Since the number of columns in the first matrix is equal to the number of rows in the second matrix (C1 = R2), these matrices can be multiplied as presented. The product of these matrices will have the dimensions defined by the number of rows in the first matrix by the number of columns in the second matrix: R1 × C2
As shown above, the order in which the matrices appear in the problem makes a difference.
Therefore, the Commutative Property does not apply to matrix multiplication unless the two matrices being multiplied are both the same size square matrices.
127
example 2.13a Determine whether these matrices can be multiplied as presented:
1 2 3 4 1 2 3
5 6 7 8 4 5 6
× 9 10 11 12 7 8 9
13 14 15 16 10 11 12
17 18 19 20
1. Determine the matrix orders: [R1 × C1] × [R2 × C2] = [4 × 3] × [5 × 4]
2. Compare C1 and R2…Are they equal or not equal???
3. Since C1 ≠ R2, these matrices cannot be multiplied as presented.
example 2.13b Determine whether these matrices can be multiplied as presented:
1 2 3 4
1 2 3 5 6 7 8
4 5 6 9 10 11 12 ×
7 8 9 13 14 15 16
10 11 12 17 18 19 20
1. Determine the matrix orders: [R1 × C1] × [R2 × C2] = [5 × 4] × [4 × 3]
2. Compare C1 and R2…Are they equal or not equal???
3. Since C1 = R2, these matrices can be multiplied as presented.
NOTE: This product will be a 5 × 3 matrix.
example 2.13c Determine whether these matrices can be multiplied as presented:
1 2 3 1 2 3 4
4 5 6 5 6 7 8 ×
7 8 9 9 10 11 12
10 11 12
1. Determine the matrix orders: [R1 × C1] × [R2 × C2] = [3 × 4] × [4 × 3]
2. Compare C1 and R2…Are they equal or not equal???
3. Since C1 = R2, these matrices can be multiplied as presented.
NOTE: This product will be a 3 × 3 matrix.
example 2.13d Determine whether these matrices can be multiplied as presented:
1 2 3
1 2 3 4 4 5 6
× 5 6 7 8 7 8 9
9 10 11 12 10 11 12
1. Determine the matrix orders: [R1 × C1] × [R2 × C2] = [4 × 3] × [3 × 4]
2. Compare C1 and R2…Are they equal or not equal???
3. Since C1 = R2, these matrices can be multiplied as presented.
NOTE: This product will be a 4 × 4 matrix.
example 2.13e Find this matrix product:
1 2 3 1 2 3 4
4 5 6 5 6 7 8 ×
7 8 9 9 10 11 12
10 11 12 1. Determine the matrix orders: [R1 × C1] × [R2 × C2] = [3 × 4] × [4 × 3]
2. Compare C1 and R2…C1 = R2…OK to multiply.
3. Determine the order of the product: [3 × 3]
www.sylvania.com
We do not reinvent the wheel we reinvent light.
Fascinating lighting offers an infinite spectrum of possibilities: Innovative technologies and new markets provide both opportunities and challenges.
An environment in which your expertise is in high demand. Enjoy the supportive working atmosphere within our global group and benefit from international career paths. Implement sustainable ideas in close cooperation with other specialists and contribute to influencing our future. Come and join us in reinventing light every day.
Light is OSRAM
129
4. Multiply the entries in the Nth row of the first matrix by the respective entries in the Nth column of the second matrix, entering the results in the appropriate cells.
Generally:
n p q
a b c d an br cu dx ap bs cv dy aq bt cw dz r s t
e f g h en fr gu hx ep fs gv hy eq ft gw hz u v w
i j k m in jr ku mx ip js kv my iq jt kw mz x y z
+ + + + + + + + +
× = + + + + + + + + +
+ + + + + + + + +
Specifically:
1(1) 2(4) 3(7) 4(10) 70 1(2) 2(5) 3(8) 4(11) 80 1(3) 2(6) 3(9) 4(12) 90 5(1) 6(4) 7(7) 8(10) 158 5(2) 6(5) 7(8) 8(11) 184 5(3) 6(6) 7(9) 8(12) 210 9(1) 10(4) 11(7) 12(10) 246 9(2) 10(5) 11(8) 12(11) 288 9(3)
+ + + = + + + = + + + =
+ + + = + + + = + + + =
+ + + = + + + = +10(6) 11(9) 12(12) 330
+ + =
requested product:
70 80 90 158 184 210 246 288 330
example 2.13f Find this matrix product:
1 2 3
1 2 3 4 4 5 6
× 5 6 7 8 7 8 9
9 10 11 12 10 11 12
1. Determine the matrix orders: [R1 × C1] × [R2 × C2] = [4 × 3] × [3 × 4]
2. Compare C1 and R2…C1 = R2…OK to multiply.
3. Determine the order of the product: [4 × 4]
4. Multiply the entries in the Nth row of the first matrix by the respective entries in the Nth column of the second matrix, entering the results in the appropriate cells:
Generally:
×
n p q na pe qi nb pf qj nc pg qk nd ph qm a b c d
r s t ra se ti rb sf tj rc sg tk rd sh tm e f g h
u v w ua ve wi ub vf wj uc vg wk ud vh wm i j k m
x y z xa ye zi xb yf zj xc yg zk xd yh zm
+ + + + + + + +
+ + + + + + + +
=
+ + + + + + + +
+ + + + + + + +
Specifically:
1(1) 2(5) 3(9) 38 1(2) 2(6) 3(10) 44 1(3) 2(7) 3(11) 50 1(4) 2(8) 3(12) 56 4(1) 5(5) 6(9) 83 4(2) 5(6) 6(10) 98 4(3) 5(7) 6(11) 113 4(4) 5(8) 6(12) 128 7(1) 8(5) 9(9) 128 7(2) 8(6) 9(10) 152 7(3) 8(7) 9(11) 176
+ + = + + = + + = + + =
+ + = + + = + + = + + =
+ + = + + = + + = 7(4) 8(8) 9(12) 200
10(1) 11(5) 12(9) 173 10(2) 11(6) 12(10) 206 10(3) 11(7) 12(11) 239 10(4) 11(8) 12(12) 272
+ + =
+ + = + + = + + = + + =
requested product:
38 44 50 56 83 98 113 128 128 152 176 200 173 206 239 272
Notice that the given matrices in example 2.13c have been reversed to create example 2.13d. As seen in example 2.13e and example 2.13f, the reversal of the matrices will produce a very different answer. Since both of these presentations can be multiplied and matrix multiplication is not normally commutative, it is very important that we put the desired matrices into the proper order to achieve the desired product.
If these matrices were derived from a real-world application, we would definitely need to pay attention to the order in which the matrix multiplication is presented.
360° thinking .
© Deloitte & Touche LLP and affiliated entities.
360° thinking .
Discover the truth at www.deloitte.ca/careers
360° thinking .
Discover the truth at www.deloitte.ca/careers
© Deloitte & Touche LLP and affiliated entities.
360° thinking .
Discover the truth at www.deloitte.ca/careers
131