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What is the Transpose of a Product of Matrices?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
The transpose of a product of matrices is found by taking the transpose of each individual matrix and then multiplying them in reverse order. If you have two matrices, A and B, their product is AB. The transpose of this product, (AB)^T, is equal to B^T A^T.
Simple Example
Quick Example
Imagine you have two teams, Team A and Team B, playing cricket. You want to record their scores over two matches. If you make a matrix for Team A's scores and another for Team B's scores, multiplying them in a certain way gives you a combined performance. Now, if you want to 'flip' this combined performance (take its transpose), you first 'flip' Team B's scores, then 'flip' Team A's scores, and then multiply these flipped scores, but in the opposite order (B's flipped scores first, then A's).
Worked Example
Step-by-Step
Let's find (AB)^T for A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]].
Step 1: First, calculate the product AB.
AB = [[1, 2], [3, 4]] * [[5, 6], [7, 8]]
AB = [[(1*5)+(2*7), (1*6)+(2*8)], [(3*5)+(4*7), (3*6)+(4*8)]]
AB = [[5+14, 6+16], [15+28, 18+32]]
AB = [[19, 22], [43, 50]]
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Step 2: Now, find the transpose of AB, which is (AB)^T.
(AB)^T = [[19, 43], [22, 50]]
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Step 3: Next, find the transpose of B, which is B^T.
B^T = [[5, 7], [6, 8]]
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Step 4: Find the transpose of A, which is A^T.
A^T = [[1, 3], [2, 4]]
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Step 5: Finally, calculate B^T A^T.
B^T A^T = [[5, 7], [6, 8]] * [[1, 3], [2, 4]]
B^T A^T = [[(5*1)+(7*2), (5*3)+(7*4)], [(6*1)+(8*2), (6*3)+(8*4)]]
B^T A^T = [[5+14, 15+28], [6+16, 18+32]]
B^T A^T = [[19, 43], [22, 50]]
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Step 6: Compare (AB)^T and B^T A^T. They are equal.
Answer: (AB)^T = [[19, 43], [22, 50]] and B^T A^T = [[19, 43], [22, 50]]. They are the same.
Why It Matters
Understanding the transpose of a product of matrices is super important in fields like AI/ML and Physics. For example, in AI, it helps train smart models that recognise faces or understand speech, and in Physics, it's used to describe how objects move and rotate. Engineers designing electric vehicles (EVs) or rockets for space technology also use this concept to make calculations for stability and control.
Common Mistakes
MISTAKE: Students often calculate (AB)^T as A^T B^T. | CORRECTION: Remember the 'reverse order' rule: (AB)^T is always B^T A^T, not A^T B^T.
MISTAKE: Forgetting to transpose the individual matrices before multiplying them in reverse order. | CORRECTION: First, find A^T and B^T, and only then multiply them as B^T A^T.
MISTAKE: Making calculation errors during matrix multiplication or transposition. | CORRECTION: Double-check each step – multiplying rows by columns for products, and swapping rows and columns for transposes.
Practice Questions
Try It Yourself
QUESTION: If A = [[1, 0], [0, 1]] and B = [[2, 3], [4, 5]], find (AB)^T. | ANSWER: [[2, 4], [3, 5]]
QUESTION: Given P = [[-1, 2], [3, 0]] and Q = [[4, 1], [0, 5]], verify that (PQ)^T = Q^T P^T. | ANSWER: Both sides will be [[-4, 12], [7, 3]].
QUESTION: Let X = [[1], [2]] and Y = [[3, 4]]. Calculate (XY)^T and X^T Y^T. Are they equal? | ANSWER: (XY)^T = [[3, 6], [4, 8]]. X^T Y^T is not possible because the number of columns in X^T (1) does not match the number of rows in Y^T (2). This shows the rule only applies when the individual transposes can be multiplied.
MCQ
Quick Quiz
Which of the following is the correct formula for the transpose of a product of two matrices, A and B?
(AB)^T = A^T B^T
(AB)^T = B^T A^T
(AB)^T = AB
(AB)^T = BA
The Correct Answer Is:
B
The correct formula for the transpose of a product of two matrices is (AB)^T = B^T A^T. The order of multiplication is reversed after taking the transpose of each matrix.
Real World Connection
In the Real World
Imagine a company like Zepto or Swiggy planning delivery routes for their riders across different neighbourhoods. They might use matrices to represent distances and traffic conditions. When they need to 'reverse' or 're-optimize' these routes (like finding the best return path), they use the concept of matrix transpose. This helps them quickly calculate efficient routes, ensuring your food or groceries arrive fast!
Key Vocabulary
Key Terms
MATRIX: A rectangular arrangement of numbers or expressions in rows and columns | TRANSPOSE: A new matrix formed by swapping the rows and columns of the original matrix | PRODUCT OF MATRICES: The result of multiplying two matrices, where rows of the first matrix are multiplied by columns of the second | ORDER OF MULTIPLICATION: The sequence in which matrices are multiplied, which affects the result (AB is generally not equal to BA)
What's Next
What to Learn Next
Now that you understand the transpose of a product, you're ready to explore how transposes are used in solving systems of linear equations. This concept is a building block for more advanced topics like inverse matrices and determinants, which are super useful in many applications!
