Matrix – Represent a Figure Using Matrices

Representing a Figure Using Matrices: 

To represent a figure using a matrix, write the x-coordinates in the first row of the matrix and write the y-coordinates in the second row of the matrix.          

Addition and Subtraction With Matrices: 

To add or subtract matrices, you add or subtract corresponding elements. The matrices must have the same dimensions. 

Example: 1 

Add the matrices:

[0234]+[1532]0324+1352

Solution: 

[0234]+ [1532]= [0+32+53+34+2]= [1766]0324+ 1352= 0+33+32+54+2= 1676

Example: 2 

Add the matrices:

[1−543−20]+[462−579]14−2−530+4276−59

Solution: 

[1−543−20]+[462−579]= [1+4−5+64+23+(−5)−2+70+9]= [516−259]14−2−530+4276−59= 1+44+2−2+7−5+63+(−5)0+9= 5651−29

Example: 3 

Subtract the matrices:

⎡⎣⎢83−7253⎤⎦⎥−⎡⎣⎢2−179−60⎤⎦⎥8235−73−29−1−670

Solution: 

⎡⎣⎢83−7253⎤⎦⎥−⎡⎣⎢2−179−60⎤⎦⎥=⎡⎣⎢8−23−(−1)−7−72−95−(−6)3−0⎤⎦⎥= ⎡⎣⎢64−14−7113⎤⎦⎥8235−73−29−1−670=8−22−93−−15−−6−7−73−0= 6−7411−143

Example: 4 

Subtract the matrices:

[1425]−[3143]1245−3413

Solution: 

[1425]−[3143]= [1−34−12−45−3]=[−23−22]1245−3413= 1−32−44−15−3=−2−232

Represent a Translation Using Matrices: 

You can use matrix addition to represent a translation in the coordinate plane. The image matrix for a translation is the sum of the translation matrix and the matrix that represents the preimage. 

Example: 1 

The matrix

[11503−1]represents ΔABC. Find the image matrix that represents the translation of ΔABC 1 unit left and 3 units up. Then graph ΔABC and its image15310-1 represents ∆ABC. Find the image matrix that represents the translation of ∆ABC 1 unit left and 3 units up. Then graph ∆ABC and its image

Example: 2

Find the image matrix that represents the translation of the polygon.

[84736521], 4 units left and 2 units down.87654321, 4 units left and 2 units down.

Solution: 

The translation matrix will be

−4−4−4−4−2−2−2−2

Add this translation matrix to the matrix with the preimage to find the image matrix 

[84736521] + [−4−2−4−2−4−4−2−2]=[8−44−27−43−26−45−42−21−2] = [4231210−1]87654321 + −4−4−4−4−2−2−2−2=8−47−46−45−44−23−22−21−2 = 4321210−1

Multiplying matrices: 

The product of two matrices, A and B, is defined only when the number of columns in A is equal to the number of rows in B. If A is an m x n matrix and B is an n x p matrix, then the product AB is an m x p matrix. 

Example: 1 

Multiply

[1405][2−1−38]10452−3−18

Solution: 

The matrices are both 2 x 2, so their product is defined. Use the following steps to find the elements of the product matrix. 

Step 1: Multiply the numbers in the first row of the first matrix by the numbers in the first column of the second matrix. Put the result in the first row, the first column of the product matrix. 

Step 2:  Multiply the numbers in the first row of the first matrix by the numbers in the second column of the second matrix. Put the result in the first row, second column of the product matrix. 

Step 3: Multiply the numbers in the second row of the first matrix by the numbers in the first column of the second matrix. Put the result in the second row, the first column of the product matrix. 

Step 4: Multiply the numbers in the second row of the first matrix by the numbers in the second column of the second matrix. Put the result in the second row, second column of the product matrix.

Step 5: Simplify the product matrix. 

Example: 2  

Multiply:

P= [1324], Q= [−1−3−2−4]P= 1234, Q= −1−2−3−4

Solution: 

Matrix “P” dimensions = 2 x 2 

Matrix “Q” dimensions = 2 x 2 

For product AB = 2 x 2 . 2 x 2 à product is defined 

[1324][−1−3−2−4]= [1 ×(−1)+2 ×(−3)3 ×(−1)+4 ×(−3)1×(−2)+2×(−4)3×(−2)+4 × (−4)]1234−1−2−3−4= 1 ×−1+2 ×−31×−2+2×−43 ×−1+4 ×−33×−2+4 × −4

=[−1−6−3−12−2−8−6−16]=[−7−15−10−22]=−1−6−2−8−3−12−6−16=−7−10−15−22

Summary

• A matrix is a rectangular arrangement of numbers in rows and columns.

(The plural of matrix is matrices.) Each number in a matrix is called an element.

• The dimensions of a matrix are the numbers of rows and columns.
• To add or subtract matrices, you add or subtract corresponding elements. The matrices must have the same dimensions.

The product of two matrices, A and B, is defined only when the number of columns in A is equal to the number of rows in B. If A is an m x n matrix and B is an n x p matrix, then the product AB is an m x p matrix.

Concept Map:

What We Have Learned:

• Define matrix, Dimensions of a matrix
• Represent figures using matrices
• Addition and subtraction of matrices
• Representing a translation using matrices
• Solve a real-world problem