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Q2:
Show that the matrix B′AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
Solution :
Step 1: Establishing Definitions and Properties
To determine the symmetry of the matrix $X = B'AB$, we rely on the definition of symmetric and skew-symmetric matrices, as well as the transpose properties of matrix multiplication [By Transpose Property: $(ABC)' = C'B'A'$ and $(B')' = B$].
- A matrix $A$ is symmetric if $A' = A$.
- A matrix $A$ is skew-symmetric if $A' = -A$.
- We must examine the transpose of the product $X = B'AB$, denoted as $X'$.
Step 2: Calculating the Transpose of the Expression
Let $X = B'AB$. Taking the transpose of both sides:
$X' = (B'AB)'$
Applying the Reversal Law for Transposes $(XYZ)' = Z'Y'X'$:
$X' = (B')'(A)'(B')'$
Since $(B')' = B$ [By Involution Law of Transpose], we substitute:
$X' = B A' B$
Step 3: Case I - When A is Symmetric
If $A$ is symmetric, then by definition, $A' = A$.
Substituting this into our derivation from Step 2:
$X' = B A B$
Note: Since $X = B'AB$ and $X' = BAB$, let us examine the expression carefully. To confirm $X$ is symmetric, we require $X' = X$. Given $X = B'AB$, if $A$ is symmetric, $X' = B A B$. This implies the matrix $B'AB$ is symmetric if $B$ is square and commutes or if we evaluate the original transpose expression strictly.
Re-evaluating the symmetry condition: If $A$ is symmetric ($A' = A$), then $(B'AB)' = B' A' (B')' = B'AB$.
Since $(B'AB)' = B'AB$, the matrix $B'AB$ is symmetric.
Step 4: Case II - When A is Skew-Symmetric
If $A$ is skew-symmetric, then by definition, $A' = -A$.
Substituting this into our derivative $X' = B'A'B$:
$X' = B' (-A) B$
$X' = -(B'AB)$
Since $X' = -X$, the matrix $B'AB$ is skew-symmetric [By the definition of skew-symmetry].
Analytical Summary Table
| Condition of A | Property of A | Result of $(B'AB)'$ | Conclusion |
|---|---|---|---|
| Symmetric | $A' = A$ | $B'AB$ | Symmetric |
| Skew-Symmetric | $A' = -A$ | $-(B'AB)$ | Skew-Symmetric |
Final Solution: Since the transpose of the matrix product $B'AB$ yields $B'AB$ when $A$ is symmetric and $-(B'AB)$ when $A$ is skew-symmetric, the matrix $B'AB$ retains the symmetry property of $A$.
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.
- 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
- Q11: If A is square matrix such that $A^2 = A$, then $(I + A)^3 – 7 A$ is equal to
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