Proof of (AB)^-1 = B^-1 A^-1 | Simple Explanation for Class 12

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What is the Proof of (AB)^-1 = B^-1 A^-1?

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

The proof of (AB)^-1 = B^-1 A^-1 shows that the inverse of a product of two matrices (A and B) is equal to the product of their individual inverses, but in reverse order. This property is fundamental in matrix algebra and helps simplify complex calculations involving matrix inverses.

Simple Example

Quick Example

Imagine you have two machines in a chai shop. Machine A makes the chai mix, and Machine B adds milk and sugar. If you want to reverse the entire process (undo the chai making), you first need to 'undo' what Machine B did (remove milk/sugar), and THEN 'undo' what Machine A did (separate the chai mix). You can't undo Machine A first if Machine B's work is still there! This is similar to how B^-1 A^-1 works for matrices.

Worked Example

Step-by-Step

Let's prove (AB)^-1 = B^-1 A^-1 using a property: for any invertible matrix M, M * M^-1 = I (Identity Matrix).

Step 1: We want to show that (B^-1 A^-1) is the inverse of (AB). This means (AB) * (B^-1 A^-1) should equal the Identity Matrix (I).
---Step 2: Start with the expression: (AB) * (B^-1 A^-1)
---Step 3: Using the associative property of matrix multiplication, we can group terms: A * (B * B^-1) * A^-1
---Step 4: We know that B * B^-1 = I (Identity Matrix). So, substitute I into the expression: A * I * A^-1
---Step 5: Multiplying any matrix by the Identity Matrix leaves the matrix unchanged. So, A * I = A. The expression becomes: A * A^-1
---Step 6: We also know that A * A^-1 = I (Identity Matrix).
---Step 7: Since (AB) * (B^-1 A^-1) = I, it means that B^-1 A^-1 is indeed the inverse of AB. Therefore, (AB)^-1 = B^-1 A^-1 is proven.

Answer: The proof shows that (AB) * (B^-1 A^-1) = I, confirming that (AB)^-1 = B^-1 A^-1.

Why It Matters

This concept is crucial in fields like AI/ML for optimizing algorithms and in physics for solving complex systems of equations. Engineers use it to design stable structures and control systems, while in FinTech, it helps manage large datasets for financial modeling. Understanding this property can open doors to careers in data science, engineering, and research.

Common Mistakes

MISTAKE: Assuming (AB)^-1 = A^-1 B^-1 | CORRECTION: Remember the order reverses! It's (AB)^-1 = B^-1 A^-1. Think of putting on socks then shoes; to reverse, you take off shoes then socks.

MISTAKE: Forgetting that A and B must be invertible matrices for their inverses to exist. | CORRECTION: Always check if the determinant of A and B is non-zero. If det(A)=0 or det(B)=0, then A^-1 or B^-1 does not exist.

MISTAKE: Confusing matrix inverse with scalar inverse. | CORRECTION: Matrix inverse is not 1/A. It's a special matrix A^-1 such that A * A^-1 = I. Matrix multiplication is not commutative, so order matters.

Practice Questions

Try It Yourself

QUESTION: If A and B are invertible matrices, which of the following is true: (AB)^-1 = ? | ANSWER: B^-1 A^-1

QUESTION: If P and Q are 3x3 invertible matrices, and (PQ)^-1 = X, what is X in terms of P^-1 and Q^-1? | ANSWER: X = Q^-1 P^-1

QUESTION: Given matrices A and B are invertible. Prove that (ABC)^-1 = C^-1 B^-1 A^-1. (Hint: Treat AB as a single matrix initially) | ANSWER: Let X = AB. Then (ABC)^-1 = (XC)^-1 = C^-1 X^-1. Substitute X = AB back: C^-1 (AB)^-1. We know (AB)^-1 = B^-1 A^-1. So, (ABC)^-1 = C^-1 B^-1 A^-1.

MCQ

Quick Quiz

Which property is used in the proof of (AB)^-1 = B^-1 A^-1?

Commutative property of matrix multiplication

Distributive property of matrix multiplication

Associative property of matrix multiplication

Scalar multiplication property

The Correct Answer Is:

The associative property (A(BC) = (AB)C) allows us to regroup the matrices as A * (B * B^-1) * A^-1, which is key to simplifying the expression to I. The other properties are not directly applied in this specific proof step.

Real World Connection

In the Real World

In computer graphics, when you apply multiple transformations (like rotating and then scaling) to an image on your mobile phone, these are represented by matrices. If you want to 'undo' these transformations to get the original image, you'd apply the inverse transformations in reverse order, just like (AB)^-1 = B^-1 A^-1. This ensures your selfies always look perfect!

Key Vocabulary

Key Terms

MATRIX: A rectangular array of numbers or functions | INVERSE MATRIX: A matrix that, when multiplied by the original matrix, gives the Identity Matrix (I) | IDENTITY MATRIX (I): A square matrix with ones on the main diagonal and zeros elsewhere, acting like the number '1' in multiplication | ASSOCIATIVE PROPERTY: A property where the grouping of terms in an operation does not change the result, like (a*b)*c = a*(b*c) for matrices.

What's Next

What to Learn Next

Now that you understand the inverse of a product, you can explore how to find the inverse of a 3x3 matrix using adjoint and determinant. This will help you solve more complex systems of equations, which are common in engineering problems.