Question

Matrix a with row p,0 and 0,2 matrix b with row 0,-q and 1,0 matrix c with rows 2,-2 and 2,2 and ba=cc then find p and q

Answer 1

p=8, q=4

Answer 2

To find the values of $p$ and $q$, we need to perform the matrix multiplications $BA$ and $CC$ and then equate the resulting matrices.

Given matrices are: $A = \begin{pmatrix} p & 0 \\ 0 & 2 \end{pmatrix}$ $B = \begin{pmatrix} 0 & -q \\ 1 & 0 \end{pmatrix}$ $C = \begin{pmatrix} 2 & -2 \\ 2 & 2 \end{pmatrix}$

The given condition is $BA = CC$.

Step 1: Calculate $BA$

$BA = \begin{pmatrix} 0 & -q \\ 1 & 0 \end{pmatrix} \begin{pmatrix} p & 0 \\ 0 & 2 \end{pmatrix}$

$BA = \begin{pmatrix} (0)(p) + (-q)(0) & (0)(0) + (-q)(2) \\ (1)(p) + (0)(0) & (1)(0) + (0)(2) \end{pmatrix}$

$BA = \begin{pmatrix} 0 + 0 & 0 - 2q \\ p + 0 & 0 + 0 \end{pmatrix}$

$BA = \begin{pmatrix} 0 & -2q \\ p & 0 \end{pmatrix}$

Step 2: Calculate $CC$

$CC = \begin{pmatrix} 2 & -2 \\ 2 & 2 \end{pmatrix} \begin{pmatrix} 2 & -2 \\ 2 & 2 \end{pmatrix}$

$CC = \begin{pmatrix} (2)(2) + (-2)(2) & (2)(-2) + (-2)(2) \\ (2)(2) + (2)(2) & (2)(-2) + (2)(2) \end{pmatrix}$

$CC = \begin{pmatrix} 4 - 4 & -4 - 4 \\ 4 + 4 & -4 + 4 \end{pmatrix}$

$CC = \begin{pmatrix} 0 & -8 \\ 8 & 0 \end{pmatrix}$

Step 3: Equate $BA$ and $CC$

According to the problem statement, $BA = CC$:

$\begin{pmatrix} 0 & -2q \\ p & 0 \end{pmatrix} = \begin{pmatrix} 0 & -8 \\ 8 & 0 \end{pmatrix}$

For two matrices to be equal, their corresponding elements must be equal. Comparing the elements:

From the element in the first row, second column: $-2q = -8$

$2q = 8$

$q = \frac{8}{2}$

$q = 4$

From the element in the second row, first column: $p = 8$

Thus, the values of $p$ and $q$ are $8$ and $4$ respectively.