Given the matrices [tex]P=\left[\begin{array}{ccc}2 & 1 & -2 \\ 3 & -1 & 1 \\ 2 & 4 & 1\end{array}\right][/tex] and [tex]Q=\left[\begin{array}{ccc}3 & -2 & 1 \\ 2 & 1 & 1 \\ 1 & 1 & -2\end{array}\right][/tex], determine [tex]( P + Q )^2+3 Q[/tex].
Answer (1)
Calculate P + Q : Add the corresponding elements of matrices P and Q.
Calculate ( P + Q ) 2 : Multiply the matrix ( P + Q ) by itself.
Calculate 3 Q : Multiply each element of matrix Q by 3.
Calculate ( P + Q ) 2 + 3 Q : Add the corresponding elements of matrices ( P + Q ) 2 and 3 Q . The final result is 26 37 40 − 16 8 − 5 − 3 − 4 2 .
Explanation
Problem Analysis We are given two matrices, P and Q , and we need to compute ( P + Q ) 2 + 3 Q . This involves matrix addition, matrix multiplication, and scalar multiplication.
Calculate P + Q First, we calculate P + Q by adding the corresponding elements of the matrices P and Q :
P + Q = 2 3 2 1 − 1 4 − 2 1 1 + 3 2 1 − 2 1 1 1 1 − 2 = 2 + 3 3 + 2 2 + 1 1 + ( − 2 ) − 1 + 1 4 + 1 − 2 + 1 1 + 1 1 + ( − 2 ) = 5 5 3 − 1 0 5 − 1 2 − 1
Calculate (P + Q)^2 Next, we calculate ( P + Q ) 2 by multiplying the matrix ( P + Q ) by itself:
( P + Q ) 2 = 5 5 3 − 1 0 5 − 1 2 − 1 5 5 3 − 1 0 5 − 1 2 − 1 = ( 5 ⋅ 5 + ( − 1 ) ⋅ 5 + ( − 1 ) ⋅ 3 ) ( 5 ⋅ 5 + 0 ⋅ 5 + 2 ⋅ 3 ) ( 3 ⋅ 5 + 5 ⋅ 5 + ( − 1 ) ⋅ 3 ) ( 5 ⋅ ( − 1 ) + ( − 1 ) ⋅ 0 + ( − 1 ) ⋅ 5 ) ( 5 ⋅ ( − 1 ) + 0 ⋅ 0 + 2 ⋅ 5 ) ( 3 ⋅ ( − 1 ) + 5 ⋅ 0 + ( − 1 ) ⋅ 5 ) ( 5 ⋅ ( − 1 ) + ( − 1 ) ⋅ 2 + ( − 1 ) ⋅ ( − 1 )) ( 5 ⋅ ( − 1 ) + 0 ⋅ 2 + 2 ⋅ ( − 1 )) ( 3 ⋅ ( − 1 ) + 5 ⋅ 2 + ( − 1 ) ⋅ ( − 1 )) = 25 − 5 − 3 25 + 0 + 6 15 + 25 − 3 − 5 + 0 − 5 − 5 + 0 + 10 − 3 + 0 − 5 − 5 − 2 + 1 − 5 + 0 − 2 − 3 + 10 + 1 = 17 31 37 − 10 5 − 8 − 6 − 7 8
Calculate 3Q Now, we calculate 3 Q by multiplying each element of the matrix Q by 3:
3 Q = 3 3 2 1 − 2 1 1 1 1 − 2 = 3 ⋅ 3 3 ⋅ 2 3 ⋅ 1 3 ⋅ ( − 2 ) 3 ⋅ 1 3 ⋅ 1 3 ⋅ 1 3 ⋅ 1 3 ⋅ ( − 2 ) = 9 6 3 − 6 3 3 3 3 − 6
Calculate (P + Q)^2 + 3Q Finally, we calculate ( P + Q ) 2 + 3 Q by adding the corresponding elements of the matrices ( P + Q ) 2 and 3 Q :
( P + Q ) 2 + 3 Q = 17 31 37 − 10 5 − 8 − 6 − 7 8 + 9 6 3 − 6 3 3 3 3 − 6 = 17 + 9 31 + 6 37 + 3 − 10 + ( − 6 ) 5 + 3 − 8 + 3 − 6 + 3 − 7 + 3 8 + ( − 6 ) = 26 37 40 − 16 8 − 5 − 3 − 4 2
Final Answer Therefore, ( P + Q ) 2 + 3 Q = 26 37 40 − 16 8 − 5 − 3 − 4 2 .
Examples
Matrix operations are fundamental in computer graphics for transformations like rotations, scaling, and translations of objects in 3D space. For instance, adding matrices can combine multiple transformations into one, while scalar multiplication can uniformly scale an object. The calculation we performed is similar to combining different transformations to achieve a final desired state in a graphical scene. This is also used in physics for describing systems of linear equations and their transformations.
