At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Get quick and reliable solutions to your questions from a community of experienced experts on our platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
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]
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]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.