Question

If A and B are two matrix of same order such that $A(B-I) = O, B(A-I) = O$, where $I$ and $O$ is identity and zero matrix respectively of same order as of A and $(A + B)^3 = K(A + B)$, then K is equal to

Answer 1

4

Answer 2

The problem provides two matrix equations and asks for the value of a scalar $K$.

Step 1: Simplify the given matrix equations. We are given:

1. $A(B-I) = O$
2. $B(A-I) = O$

Expanding the first equation: $AB - AI = O$ Since $AI = A$ (where $I$ is the identity matrix), we get: $AB - A = O \implies AB = A$ (Equation 1)

Expanding the second equation: $BA - BI = O$ Since $BI = B$, we get: $BA - B = O \implies BA = B$ (Equation 2)

Step 2: Determine the properties of matrices A and B using the derived relations. We will use Equation 1 ($AB=A$) and Equation 2 ($BA=B$) to find $A^2$ and $B^2$.

Consider $A^2$: $A^2 = A \cdot A$ From Equation 1, we know $A = AB$. Substitute this into the expression for $A^2$: $A^2 = (AB)A$ Using the associativity of matrix multiplication: $A^2 = A(BA)$ From Equation 2, we know $BA = B$. Substitute this into the expression for $A^2$: $A^2 = AB$ From Equation 1 again, we know $AB = A$. Therefore: $A^2 = A$ This means matrix A is idempotent.

Consider $B^2$: $B^2 = B \cdot B$ From Equation 2, we know $B = BA$. Substitute this into the expression for $B^2$: $B^2 = (BA)B$ Using the associativity of matrix multiplication: $B^2 = B(AB)$ From Equation 1, we know $AB = A$. Substitute this into the expression for $B^2$: $B^2 = BA$ From Equation 2 again, we know $BA = B$. Therefore: $B^2 = B$ This means matrix B is idempotent.

So, we have established the following properties:

• $A^2 = A$
• $B^2 = B$
• $AB = A$
• $BA = B$

Step 3: Compute $(A+B)^2$. $(A+B)^2 = (A+B)(A+B)$ $= A \cdot A + A \cdot B + B \cdot A + B \cdot B$ $= A^2 + AB + BA + B^2$ Now, substitute the properties derived in Step 2: $= A + A + B + B$ $= 2A + 2B$ $= 2(A+B)$

Step 4: Compute $(A+B)^3$. $(A+B)^3 = (A+B)^2 (A+B)$ From Step 3, we know $(A+B)^2 = 2(A+B)$. Substitute this: $(A+B)^3 = 2(A+B)(A+B)$ $= 2(A+B)^2$ Substitute $(A+B)^2 = 2(A+B)$ again: $= 2[2(A+B)]$ $= 4(A+B)$

Step 5: Determine the value of K. We are given that $(A+B)^3 = K(A+B)$. From Step 4, we found $(A+B)^3 = 4(A+B)$. Comparing the two expressions: $K(A+B) = 4(A+B)$ Assuming $A+B \neq O$ (the zero matrix), we can equate the scalar coefficients. If $A+B=O$, then $K$ can be any scalar; however, in such problems, a unique value for $K$ is expected. Therefore, $K=4$.