IDNLearn.com: Your destination for reliable and timely answers to any question. Whether your question is simple or complex, our community is here to provide detailed and trustworthy answers quickly and effectively.
Sagot :
To solve the problem of finding [tex]\(3A - 4B^{\prime}\)[/tex], we need to follow these steps:
1. Compute [tex]\(3A\)[/tex]: This involves multiplying each element of matrix [tex]\(A\)[/tex] by 3.
2. Compute [tex]\(B^{\prime}\)[/tex]: This is the transpose of matrix [tex]\(B\)[/tex], where we switch the rows and columns of [tex]\(B\)[/tex].
3. Compute [tex]\(4B^{\prime}\)[/tex]: This involves multiplying each element of the transposed matrix [tex]\(B^{\prime}\)[/tex] by 4.
4. Compute [tex]\(3A - 4B^{\prime}\)[/tex]: This is done by subtracting the corresponding elements of [tex]\(4B^{\prime}\)[/tex] from [tex]\(3A\)[/tex].
Let's go through each of these steps in detail:
### Step 1: Compute [tex]\(3A\)[/tex]
Matrix [tex]\(A\)[/tex] is:
[tex]\[ A = \begin{pmatrix} 1 & 5 & 3 \\ 2 & 4 & 0 \\ 3 & -1 & -5 \end{pmatrix} \][/tex]
Multiplying each element of [tex]\(A\)[/tex] by 3 gives:
[tex]\[ 3A = 3 \times \begin{pmatrix} 1 & 5 & 3 \\ 2 & 4 & 0 \\ 3 & -1 & -5 \end{pmatrix} = \begin{pmatrix} 3 & 15 & 9 \\ 6 & 12 & 0 \\ 9 & -3 & -15 \end{pmatrix} \][/tex]
### Step 2: Compute the transpose of [tex]\(B\)[/tex]
Matrix [tex]\(B\)[/tex] is:
[tex]\[ B = \begin{pmatrix} 2 & -1 & 0 \\ 0 & -2 & 5 \\ 1 & 2 & 0 \end{pmatrix} \][/tex]
The transpose of [tex]\(B\)[/tex], denoted as [tex]\(B^{\prime}\)[/tex], is:
[tex]\[ B^{\prime} = \begin{pmatrix} 2 & 0 & 1 \\ -1 & -2 & 2 \\ 0 & 5 & 0 \end{pmatrix} \][/tex]
### Step 3: Compute [tex]\(4B^{\prime}\)[/tex]
Multiplying each element of [tex]\(B^{\prime}\)[/tex] by 4 gives:
[tex]\[ 4B^{\prime} = 4 \times \begin{pmatrix} 2 & 0 & 1 \\ -1 & -2 & 2 \\ 0 & 5 & 0 \end{pmatrix} = \begin{pmatrix} 8 & 0 & 4 \\ -4 & -8 & 8 \\ 0 & 20 & 0 \end{pmatrix} \][/tex]
### Step 4: Compute [tex]\(3A - 4B^{\prime}\)[/tex]
Subtracting the corresponding elements of [tex]\(4B^{\prime}\)[/tex] from [tex]\(3A\)[/tex] gives:
[tex]\[ 3A - 4B^{\prime} = \begin{pmatrix} 3 & 15 & 9 \\ 6 & 12 & 0 \\ 9 & -3 & -15 \end{pmatrix} - \begin{pmatrix} 8 & 0 & 4 \\ -4 & -8 & 8 \\ 0 & 20 & 0 \end{pmatrix} = \begin{pmatrix} 3 - 8 & 15 - 0 & 9 - 4 \\ 6 - (-4) & 12 - (-8) & 0 - 8 \\ 9 - 0 & -3 - 20 & -15 - 0 \end{pmatrix} \][/tex]
Simplifying this, we get:
[tex]\[ 3A - 4B^{\prime} = \begin{pmatrix} -5 & 15 & 5 \\ 10 & 20 & -8 \\ 9 & -23 & -15 \end{pmatrix} \][/tex]
Hence, the final result is:
[tex]\[ 3A - 4B^{\prime} = \begin{pmatrix} -5 & 15 & 5 \\ 10 & 20 & -8 \\ 9 & -23 & -15 \end{pmatrix} \][/tex]
1. Compute [tex]\(3A\)[/tex]: This involves multiplying each element of matrix [tex]\(A\)[/tex] by 3.
2. Compute [tex]\(B^{\prime}\)[/tex]: This is the transpose of matrix [tex]\(B\)[/tex], where we switch the rows and columns of [tex]\(B\)[/tex].
3. Compute [tex]\(4B^{\prime}\)[/tex]: This involves multiplying each element of the transposed matrix [tex]\(B^{\prime}\)[/tex] by 4.
4. Compute [tex]\(3A - 4B^{\prime}\)[/tex]: This is done by subtracting the corresponding elements of [tex]\(4B^{\prime}\)[/tex] from [tex]\(3A\)[/tex].
Let's go through each of these steps in detail:
### Step 1: Compute [tex]\(3A\)[/tex]
Matrix [tex]\(A\)[/tex] is:
[tex]\[ A = \begin{pmatrix} 1 & 5 & 3 \\ 2 & 4 & 0 \\ 3 & -1 & -5 \end{pmatrix} \][/tex]
Multiplying each element of [tex]\(A\)[/tex] by 3 gives:
[tex]\[ 3A = 3 \times \begin{pmatrix} 1 & 5 & 3 \\ 2 & 4 & 0 \\ 3 & -1 & -5 \end{pmatrix} = \begin{pmatrix} 3 & 15 & 9 \\ 6 & 12 & 0 \\ 9 & -3 & -15 \end{pmatrix} \][/tex]
### Step 2: Compute the transpose of [tex]\(B\)[/tex]
Matrix [tex]\(B\)[/tex] is:
[tex]\[ B = \begin{pmatrix} 2 & -1 & 0 \\ 0 & -2 & 5 \\ 1 & 2 & 0 \end{pmatrix} \][/tex]
The transpose of [tex]\(B\)[/tex], denoted as [tex]\(B^{\prime}\)[/tex], is:
[tex]\[ B^{\prime} = \begin{pmatrix} 2 & 0 & 1 \\ -1 & -2 & 2 \\ 0 & 5 & 0 \end{pmatrix} \][/tex]
### Step 3: Compute [tex]\(4B^{\prime}\)[/tex]
Multiplying each element of [tex]\(B^{\prime}\)[/tex] by 4 gives:
[tex]\[ 4B^{\prime} = 4 \times \begin{pmatrix} 2 & 0 & 1 \\ -1 & -2 & 2 \\ 0 & 5 & 0 \end{pmatrix} = \begin{pmatrix} 8 & 0 & 4 \\ -4 & -8 & 8 \\ 0 & 20 & 0 \end{pmatrix} \][/tex]
### Step 4: Compute [tex]\(3A - 4B^{\prime}\)[/tex]
Subtracting the corresponding elements of [tex]\(4B^{\prime}\)[/tex] from [tex]\(3A\)[/tex] gives:
[tex]\[ 3A - 4B^{\prime} = \begin{pmatrix} 3 & 15 & 9 \\ 6 & 12 & 0 \\ 9 & -3 & -15 \end{pmatrix} - \begin{pmatrix} 8 & 0 & 4 \\ -4 & -8 & 8 \\ 0 & 20 & 0 \end{pmatrix} = \begin{pmatrix} 3 - 8 & 15 - 0 & 9 - 4 \\ 6 - (-4) & 12 - (-8) & 0 - 8 \\ 9 - 0 & -3 - 20 & -15 - 0 \end{pmatrix} \][/tex]
Simplifying this, we get:
[tex]\[ 3A - 4B^{\prime} = \begin{pmatrix} -5 & 15 & 5 \\ 10 & 20 & -8 \\ 9 & -23 & -15 \end{pmatrix} \][/tex]
Hence, the final result is:
[tex]\[ 3A - 4B^{\prime} = \begin{pmatrix} -5 & 15 & 5 \\ 10 & 20 & -8 \\ 9 & -23 & -15 \end{pmatrix} \][/tex]
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Accurate answers are just a click away at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.
