Exercises for Question Type 2: Determining validity of matrix operations

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Exercises for Question Type 2: Determining validity of matrix operations - IB

Question 1

Verification of the associative property of matrix multiplication using dimension analysis and entry-wise summation notation.

Let A be an m \times n matrix, B be an n \times p matrix, and C be a p \times q matrix. Show that matrix multiplication is associative by comparing (A B) C and A (B C) .

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Question 2

Let A, B, C be all 2 \times 2 matrices. Determine whether the equality A (B + C) = A B + A C holds for all such matrices. Justify your answer.

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Question 3

Let A be an invertible 2 \times 2 matrix. Show that (A^{T})^{- 1} = (A^{- 1})^{T} .

Show that (A^{T})^{- 1} = (A^{- 1})^{T} .

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Question 4

Matrices

Let A be a 2 \times 3 matrix and B be a 3 \times 4 matrix. Determine whether the product A B is defined. If it is, state the dimensions of A B .

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Question 5

Given A = (1324), B = (5768), compute A B and B A, then determine whether A B = B A .

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Question 6

Let A and B be two 2 \times 2 matrices. Determine whether in general A B = B A . Provide a justification.

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Question 7

Let A and B be invertible n \times n matrices. Determine the formula for (A B)^{- 1} in terms of A^{- 1} and B^{- 1} .

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Question 8

Let A be a 2 \times 3 matrix, B a 3 \times 4 matrix, and C a 4 \times 2 matrix. Determine whether (A B) C and A (B C) are defined, and state their dimensions.

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Question 9

Let A = (102) (1 \times 3), B = 123 (3 \times 1) . Compute A B, and determine whether B A is defined.

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Question 10

Let A = (1224) . Determine whether A^{- 1} exists. Explain your reasoning.

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Question 11

Let A be a 2 \times 3 matrix and B be a 2 \times 3 matrix. Determine whether the product A B is defined. If not, explain why.

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Question 12

Let A be a 2 \times 3 matrix and B be a 3 \times 2 matrix. Determine whether the sum A + B is defined. Explain .

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IB Mathematics Applications & Interpretation Question Type 2: Determining Validity of Matrix Operations Exercises