Math

QuestionCompute (i) A(B+C)A(B+C) and (ii) (B+C)A(B+C)A for matrices A=(1234)A=\begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix}, B=(2142)B=\begin{pmatrix}2 & 1 \\ 4 & 2\end{pmatrix}, C=(5174)C=\begin{pmatrix}5 & 1 \\ 7 & 4\end{pmatrix}.

Studdy Solution

STEP 1

Assumptions1. The matrices A, B, and C are given as follows A=[134],B=[14],C=[5174]A=\left[\begin{array}{ll}1 & \\3 &4\end{array}\right], B=\left[\begin{array}{ll} &1 \\4 &\end{array}\right], C=\left[\begin{array}{ll}5 &1 \\7 &4\end{array}\right]. The operations of matrix addition and multiplication are defined as per standard linear algebra.

STEP 2

First, we need to compute the sum of matrices B and C.
B+C=[2142]+[5174]B + C = \left[\begin{array}{ll}2 &1 \\4 &2\end{array}\right] + \left[\begin{array}{ll}5 &1 \\7 &4\end{array}\right]

STEP 3

Perform the matrix addition operation. Each element in the resulting matrix is the sum of the corresponding elements in matrices B and C.
B+C=[2+51+1+72+]B + C = \left[\begin{array}{ll}2+5 &1+1 \\+7 &2+\end{array}\right]

STEP 4

Calculate the result of the matrix addition.
B+C=[72116]B + C = \left[\begin{array}{ll}7 &2 \\11 &6\end{array}\right]

STEP 5

Now, we need to compute the product of matrix A and the sum of matrices B and C.A(B+C)=[1234][7211]A(B + C) = \left[\begin{array}{ll}1 &2 \\3 &4\end{array}\right] \left[\begin{array}{ll}7 &2 \\11 &\end{array}\right]

STEP 6

Perform the matrix multiplication operation. Each element in the resulting matrix is the dot product of the corresponding row of the first matrix and column of the second matrix.
A(B+C)=[1+21112+263+41132+46]A(B + C) = \left[\begin{array}{ll}1*+2*11 &1*2+2*6 \\3*+4*11 &3*2+4*6\end{array}\right]

STEP 7

Calculate the result of the matrix multiplication.
A(B+C)=[29147130]A(B + C) = \left[\begin{array}{ll}29 &14 \\71 &30\end{array}\right]

STEP 8

Now, we need to compute the product of the sum of matrices B and C and matrix A.(B+C)A=[72116][1234](B + C)A = \left[\begin{array}{ll}7 &2 \\11 &6\end{array}\right] \left[\begin{array}{ll}1 &2 \\3 &4\end{array}\right]

STEP 9

Perform the matrix multiplication operation. Each element in the resulting matrix is the dot product of the corresponding row of the first matrix and column of the second matrix.
(B+C)A=[7+2372+2411+63112+64](B + C)A = \left[\begin{array}{ll}7*+2*3 &7*2+2*4 \\11*+6*3 &11*2+6*4\end{array}\right]

STEP 10

Calculate the result of the matrix multiplication.
(B+C)A=[13182938](B + C)A = \left[\begin{array}{ll}13 &18 \\29 &38\end{array}\right]So, the solutions are(i) A(B+C)=[29147130]A(B + C) = \left[\begin{array}{ll}29 &14 \\71 &30\end{array}\right](ii) (B+C)A=[13182938](B + C)A = \left[\begin{array}{ll}13 &18 \\29 &38\end{array}\right]

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord