4-2 Multiplying Matrices Warm Up Lesson Presentation Lesson Quiz

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4-2 Multiplying Matrices Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

Warm Up 1. What are the dimensions of A? 2. What is entry D12? Evaluate if possible. 3. 2A — C 4. C + 2D 5. 10(2B + D) 3  2 –2 Not possible

Objectives Understand the properties of matrices with respect to multiplication. Multiply two matrices.

Vocabulary matrix product square matrix main diagonal multiplicative identity matrix

An m  n matrix A can be identified by using the notation Am  n.

Example 1A: Identifying Matrix Products Tell whether the product is defined. If so, give its dimensions. A3  4 and B4  2; AB A B AB 3  4 4  2 = 3  2 matrix The inner dimensions are equal (4 = 4), so the matrix product is defined. The dimensions of the product are the outer numbers, 3  2.

Example 1B: Identifying Matrix Products Tell whether the product is defined. If so, give its dimensions. C1  4 and D3  4; CD C D 1  4 3  4 The inner dimensions are not equal (4 ≠ 3), so the matrix product is not defined.

Check It Out! Example 1c Tell whether the product is defined. If so, give its dimensions. P2  5 Q5  3 R4  3 S4  5 SQ S Q 4  5 5  3 The inner dimensions are equal (5 = 5), so the matrix product is defined. The dimensions of the product are the outer numbers, 4  3.

Example 2A: Finding the Matrix Product Find the product, if possible. WX Check the dimensions. W is 3  2 , X is 2  3 . WX is defined and is 3  3.

Example 2A Continued Multiply row 1 of W and column 1 of X as shown. Place the result in wx11. 3(4) + –2(5)

Example 2A Continued Multiply row 1 of W and column 2 of X as shown. Place the result in wx12. 3(7) + –2(1)

Example 2A Continued Multiply row 1 of W and column 3 of X as shown. Place the result in wx13. 3(–2) + –2(–1)

Example 2A Continued Multiply row 2 of W and column 1 of X as shown. Place the result in wx21. 1(4) + 0(5)

Example 2A Continued Multiply row 2 of W and column 2 of X as shown. Place the result in wx22. 1(7) + 0(1)

Example 2A Continued Multiply row 2 of W and column 3 of X as shown. Place the result in wx23. 1(–2) + 0(–1)

Example 2A Continued Multiply row 3 of W and column 1 of X as shown. Place the result in wx31. 2(4) + –1(5)

Example 2A Continued Multiply row 3 of W and column 2 of X as shown. Place the result in wx32. 2(7) + –1(1)

Example 2A Continued Multiply row 3 of W and column 3 of X as shown. Place the result in wx33. 2(–2) + –1(–1)

Example 2B: Finding the Matrix Product Find each product, if possible. XW Check the dimensions. X is 2  3, and W is 3  2 so the product is defined and is 2  2.

Check It Out! Example 2a Find the product, if possible. BC Check the dimensions. B is 3  2, and C is 2  2 so the product is defined and is 3  2.

Lesson Quiz Evaluate if possible. 1. AB 2. BA 3. A2 4. BD 5. C3

Lesson Quiz Evaluate if possible. 1. AB 2. BA 3. A2 4. BD 5. C3 not possible not possible