To determine the matrices [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] given the matrices [tex]\( P + Q \)[/tex] and [tex]\( P - Q \)[/tex], we can utilize the following method.
We start with the two given equations:
[tex]\[ P + Q = \begin{bmatrix} 5 & 2 \\ 0 & 9 \end{bmatrix} \][/tex]
[tex]\[ P - Q = \begin{bmatrix} 3 & 6 \\ -3 & 2 \end{bmatrix} \][/tex]
To find [tex]\( P \)[/tex], we add these two equations:
[tex]\[ (P + Q) + (P - Q) = \begin{bmatrix} 5 & 2 \\ 0 & 9 \end{bmatrix} + \begin{bmatrix} 3 & 6 \\ -3 & 2 \end{bmatrix} \][/tex]
By performing the matrix addition:
[tex]\[ \begin{bmatrix} 5 & 2 \\ 0 & 9 \end{bmatrix} + \begin{bmatrix} 3 & 6 \\ -3 & 2 \end{bmatrix} = \begin{bmatrix} 5+3 & 2+6 \\ 0-3 & 9+2 \end{bmatrix} = \begin{bmatrix} 8 & 8 \\ -3 & 11 \end{bmatrix} \][/tex]
Since [tex]\( 2P = \begin{bmatrix} 8 & 8 \\ -3 & 11 \end{bmatrix} \)[/tex], we divide by 2 to obtain:
[tex]\[ P = \frac{1}{2} \begin{bmatrix} 8 & 8 \\ -3 & 11 \end{bmatrix} = \begin{bmatrix} 4 & 4 \\ -1.5 & 5.5 \end{bmatrix} \][/tex]
Next, to find [tex]\( Q \)[/tex], we subtract [tex]\( P - Q \)[/tex] from [tex]\( P + Q \)[/tex]:
[tex]\[ (P + Q) - (P - Q) = \begin{bmatrix} 5 & 2 \\ 0 & 9 \end{bmatrix} - \begin{bmatrix} 3 & 6 \\ -3 & 2 \end{bmatrix} \][/tex]
By performing the matrix subtraction:
[tex]\[ \begin{bmatrix} 5 & 2 \\ 0 & 9 \end{bmatrix} - \begin{bmatrix} 3 & 6 \\ -3 & 2 \end{bmatrix} = \begin{bmatrix} 5-3 & 2-6 \\ 0+3 & 9-2 \end{bmatrix} = \begin{bmatrix} 2 & -4 \\ 3 & 7 \end{bmatrix} \][/tex]
Since [tex]\( 2Q = \begin{bmatrix} 2 & -4 \\ 3 & 7 \end{bmatrix} \)[/tex], we divide by 2 to obtain:
[tex]\[ Q = \frac{1}{2} \begin{bmatrix} 2 & -4 \\ 3 & 7 \end{bmatrix} = \begin{bmatrix} 1 & -2 \\ 1.5 & 3.5 \end{bmatrix} \][/tex]
Thus, the matrices [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] are:
[tex]\[ P = \begin{bmatrix} 4 & 4 \\ -1.5 & 5.5 \end{bmatrix} \][/tex]
[tex]\[ Q = \begin{bmatrix} 1 & -2 \\ 1.5 & 3.5 \end{bmatrix} \][/tex]
