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Q11:
If A is square matrix such that $A^2 = A$, then $(I + A)^3 – 7 A$ is equal to
Solution :
Given Variables & Initial Setup
We are provided with a square matrix $A$ that satisfies the idempotent property:
$A^2 = A$ [Definition of Idempotent Matrix]
We are required to evaluate the expression:
$E = (I + A)^3 - 7A$
Step 1: Expanding the Binomial Expression
Since $I$ (the Identity matrix) and $A$ commute (i.e., $IA = AI = A$), we can apply the binomial theorem for the expansion of $(I + A)^3$:
$(I + A)^3 = I^3 + 3I^2A + 3IA^2 + A^3$
Simplifying the terms using properties of identity matrices ($I^n = I$ and $IA = A$):
$(I + A)^3 = I + 3A + 3A^2 + A^3$ [Property of Identity matrix multiplication]
Step 2: Utilizing the Idempotent Property
Given $A^2 = A$, we can derive higher powers of $A$:
- $A^3 = A^2 \cdot A = (A) \cdot A = A^2 = A$
- $A^n = A$ for all $n \in \mathbb{Z}^+$
Substituting these relations back into the expanded expression from Step 1:
$(I + A)^3 = I + 3A + 3(A) + (A)$
$(I + A)^3 = I + 3A + 3A + A$
$(I + A)^3 = I + 7A$
Step 3: Final Substitution and Simplification
Now, we substitute the simplified expression of $(I + A)^3$ back into the original expression $E$:
$E = (I + 7A) - 7A$
$E = I + (7A - 7A)$ [Associative property of matrix addition]
$E = I + O$ (where $O$ is the Null matrix)
$E = I$
Final Solution: (I + A)^3 – 7A = I
More Questions from Class 12 Mathematics Matrices Miscellaneous Exercise on Chapter 3
- Q1: If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.
- Q2: Show that the matrix B′AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
- Q3: Find the values of $x, y, z$ if the matrix $A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix}$ satisfy the equation $A′A = I$.
- Q4: For what values of $x$ : $\begin{bmatrix} 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 2 \\ x \end{bmatrix} = O$?
- Q5: If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$, show that $A^2 – 5A + 7I = 0$.
- Q6: Find $x$, if $\begin{bmatrix} x & -5 & -1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix} \begin{bmatrix} x \\ 4 \\ 1 \end{bmatrix} = O$
- Q7(a): A manufacturer produces three products $x, y, z$ which he sells in two markets. Annual sales are indicated below: (a) If unit sale prices of $x, y$ and $z$ are ` 2.50, ` 1.50 and ` 1.00, respectively, find the total revenue in each market with the help of matrix algebra.
- Q7(b): A manufacturer produces three products $x, y, z$ which he sells in two markets. Annual sales are indicated below: (b) If the unit costs of the above three commodities are ` 2.00, ` 1.00 and 50 paise respectively. Find the gross profit.
- Q8: Find the matrix $X$ so that $X \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix}$
- Q9: If $A = \begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}$ is such that $A^2 = I$, then
- Q10: If the matrix A is both symmetric and skew symmetric, then
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