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Adding, subtracting, and multiplying matrices
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A matrix is a rectangular array of numbers.
Example:
Add the matrices: [2 3] + [1 4] [5 1] [2 0]
Step 1: Add corresponding elements: [2+1 3+4] = [3 7] [5+2 1+0] [7 1]
Step 2: Note that matrices must have the same dimensions to be added
Answer: [3 7] [7 1]
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Dimensions: rows ร columns (this is a 2ร2 matrix)
Add corresponding entries. Matrices must have same dimensions.
Subtract corresponding entries.
Multiply every entry by the scalar:
Not commutative! in general
For and :
Entry formula: (row of ) ยท (column of )
Example:
Add corresponding entries:
Answer:
Multiply the matrix by the scalar: 3[2 -1] [4 5]
Step 1: Multiply each element by 3: 3[2 -1] = [3ยท2 3ยท(-1)] [4 5] [3ยท4 3ยท5]
Step 2: Calculate: = [6 -3] [12 15]
Answer: [6 -3] [12 15]
Multiply:
This is a 2ร2 matrix times a 2ร1 matrix. Result will be 2ร1.
First entry:
Second entry:
Multiply: [1 2][5] [3 4][6]
Step 1: Check dimensions: First matrix: 2ร2 Second matrix: 2ร1 Result will be: 2ร1
Step 2: Calculate first element (row 1 ร column): 1(5) + 2(6) = 5 + 12 = 17
Step 3: Calculate second element (row 2 ร column): 3(5) + 4(6) = 15 + 24 = 39
Step 4: Write result: [17] [39]
Answer: [17] [39]
Multiply: [1 2][5 6] [3 4][7 8]
Step 1: Check dimensions: Both are 2ร2, result will be 2ร2
Step 2: Calculate element (1,1): Row 1 ร Column 1: 1(5) + 2(7) = 5 + 14 = 19
Step 3: Calculate element (1,2): Row 1 ร Column 2: 1(6) + 2(8) = 6 + 16 = 22
Step 4: Calculate element (2,1): Row 2 ร Column 1: 3(5) + 4(7) = 15 + 28 = 43
Step 5: Calculate element (2,2): Row 2 ร Column 2: 3(6) + 4(8) = 18 + 32 = 50
Step 6: Write result: [19 22] [43 50]
Answer: [19 22] [43 50]
Multiply:
Calculate each entry of the 2ร2 result:
Entry (1,1):
Entry (1,2):
Given A = [1 2] and B = [5 6], find 2A - 3B. [3 4] [7 8]
Step 1: Calculate 2A: 2A = 2[1 2] = [2 4] [3 4] [6 8]
Step 2: Calculate 3B: 3B = 3[5 6] = [15 18] [7 8] [21 24]
Step 3: Subtract 3B from 2A: 2A - 3B = [2 4] - [15 18] [6 8] [21 24]
Step 4: Subtract corresponding elements: = [2-15 4-18] [6-21 8-24]
Step 5: Simplify: = [-13 -14] [-15 -16]
Answer: [-13 -14] [-15 -16]
Answer:
Entry (2,1):
Entry (2,2):
Answer: