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Example 1 Dimensions of Matrix Products A. Determine whether the product of A 3×4 and B 4×2 is defined. If so, state the dimensions of the product. Answer: A●B=AB 3 × 44 × 23 × 2
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Five-Minute Check (over Lesson 3–5)Then/NowExample 1: Dimensions of Matrix ProductsKey Concept: Multiplying MatricesExample 2: Multiply Square MatricesExample 3: Real-World Example: Multiply MatricesExample 4: Test of the Commutative PropertyExample 5: Test of the Distributive PropertyKey Concept: Properties of Matrix Multiplication
You multiplied matrices by a scalar.
• Multiply matrices.
• Use the properties of matrix multiplication.
Dimensions of Matrix Products
A. Determine whether the product of A3×4 and B4×2 is defined. If so, state the dimensions of the product.
Answer:
A ● B = AB
3 × 4 4 × 2 3 × 2
Dimensions of Matrix Products
A. Determine whether the product of A3×4 and B4×2 is defined. If so, state the dimensions of the product.
Answer: The inner dimensions are equal so the matrix product is defined. The dimensions of the product are 3 × 2.
A ● B = AB
3 × 4 4 × 2 3 × 2
Dimensions of Matrix Products
B. Determine whether the product of A3×2 and B4×3 is defined. If so, state the dimensions of the product.
Answer:
A ● B
3 × 2 4 × 3
Dimensions of Matrix Products
B. Determine whether the product of A3×2 and B4×3 is defined. If so, state the dimensions of the product.
Answer: The inner dimensions are not equal, so the matrix product is not defined.
A ● B
3 × 2 4 × 3
A. 3 × 3
B. 2 × 2
C. 3 × 2
D. The matrix product is not defined.
A. Determine whether the matrix product is defined. If so, what are the dimensions of the product?A3×2 and B2×3
A. 3 × 3
B. 2 × 2
C. 3 × 2
D. The matrix product is not defined.
A. Determine whether the matrix product is defined. If so, what are the dimensions of the product?A3×2 and B2×3
A. 2 × 3
B. 3 × 2
C. 2 × 2
D. The matrix product is not defined.
B. Determine whether the matrix product is defined. If so, what are the dimensions of the product?A2×3 and B2×3
A. 2 × 3
B. 3 × 2
C. 2 × 2
D. The matrix product is not defined.
B. Determine whether the matrix product is defined. If so, what are the dimensions of the product?A2×3 and B2×3
Multiply Square Matrices
Multiply Square Matrices
Step 1 Multiply the numbers in the first row of R by the numbers in the first column of S, add the products, and put the result in the first row, first column of RS.
Multiply Square Matrices
Step 2 Multiply the numbers in the first row of R by the numbers in the second column of S, add the products, and put the result in the first row, second column of RS.
Multiply Square Matrices
Step 3 Multiply the numbers in the second row of R by the numbers in the first column of S, add the products, and put the result in the second row, first column of RS.
Multiply Square Matrices
Step 4 Multiply the numbers in the second row of R by the numbers in the second column of S, add the products, and put the result in the second row, second column of RS.
Multiply Square Matrices
Step 5 Simplify the product matrix.
Answer:
Multiply Square Matrices
Step 5 Simplify the product matrix.
Answer:
A.
B.
C.
D.
A.
B.
C.
D.
Multiply Matrices
CHESS Three teams competed in the final round of
the Chess Club’s championships. For each win, a team was awarded 3 points and for each draw a team received 1 point. Which team won the tournament?
Understand The final scores can be found by multiplying the wins and draws by the points for each.
Multiply Matrices
Plan Write the results from the championship andthe points in matrix form. Set up the matricesso that the number of rows in the pointsmatrix equals the number of columns in the
results matrix.
Results Points
Multiply Matrices
Solve Multiply the matrices.
Write an equation.
Multiply columns by rows.
Multiply Matrices
Simplify.
The labels for the product matrix are shown below.
Blue
Red
Green
Total Points
Multiply Matrices
Answer:
Multiply Matrices
Answer: The red team won the championship with a total of 21 points.
Check R is a 3 × 2 matrix and P is a 2 × 1 matrix.Their product should be a 3 × 1 matrix.
A. Warton
B. Bryant
C. Chris
D. none of the above
BASKETBALL In Thursday night’s basketball game, three of the players made the points listed below in the chart. They scored 1 point for the free-throws, 2 points for the 2-point shots, and 3 points for the 3-points shots. Who scored the most points?
A. Warton
B. Bryant
C. Chris
D. none of the above
BASKETBALL In Thursday night’s basketball game, three of the players made the points listed below in the chart. They scored 1 point for the free-throws, 2 points for the 2-point shots, and 3 points for the 3-points shots. Who scored the most points?
Test of the Commutative Property
Substitution
Multiply columns by rows.
Simplify.
A. Find KL if K
Test of the Commutative Property
Answer:
Test of the Commutative Property
Answer:
Test of the Commutative Property
Substitution
Multiply columns by rows.
B. Find LK if K
Test of the Commutative Property
Answer:
Simplify.
Test of the Commutative Property
Answer:
Simplify.
A.
B.
C.
D.
A.
B.
C.
D.
A.
B.
C.
D.
A.
B.
C.
D.
Test of the Distributive Property
Substitution
Add corresponding elements.
A.
Test of the Distributive Property
Multiply columns by rows.
Answer:
Test of the Distributive Property
Multiply columns by rows.
Answer:
Test of the Distributive Property
Multiply columns by rows.
Substitution
Test of the Distributive Property
Simplify.
Answer:
Add corresponding elements.
Test of the Distributive Property
Simplify.
Answer:
Add corresponding elements.
A. B.
C. D.
A. B.
C. D.
A. B.
C. D.
A. B.
C. D.
