Let A = ( 1 3 4 1 3 2 4 10 9 ) , B = ( 7 13 -6 -1-7 2-2 2 0 ) , C = ( 1 1 2 ) ii) evaluate ( A^2 - B transpose ) C

We are tasked with evaluating \((A^2 - B^T)C\), where \(A\), \(B\), and \(C\) are defined as:

\[
A = \begin{pmatrix} 1 & 3 & 4 \\ 1 & 3 & 2 \\ 4 & 10 & 9 \end{pmatrix}, \quad
B = \begin{pmatrix} 7 & 13 & -6 \\ -1 & -7 & 2 \\ -2 & 2 & 0 \end{pmatrix}, \quad
C = \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}
\]

### Step 1: Compute \(A^2\)
To calculate \(A^2\), we multiply \(A\) by itself:

\[
A^2 = A \times A = \begin{pmatrix} 1 & 3 & 4 \\ 1 & 3 & 2 \\ 4 & 10 & 9 \end{pmatrix} \times \begin{pmatrix} 1 & 3 & 4 \\ 1 & 3 & 2 \\ 4 & 10 & 9 \end{pmatrix}
\]

Let's compute the matrix multiplication.

### Step 2: Transpose of matrix \(B\)
The transpose of matrix \(B\), denoted as \(B^T\), is obtained by switching its rows and columns.

\[
B^T = \begin{pmatrix} 7 & -1 & -2 \\ 13 & -7 & 2 \\ -6 & 2 & 0 \end{pmatrix}
\]

### Step 3: Compute \(A^2 - B^T\)
We now subtract \(B^T\) from \(A^2\).

### Step 4: Multiply \((A^2 - B^T)\) by vector \(C\)
Finally, we will multiply the resulting matrix by \(C\).

I'll calculate each of these steps to give the final result.The result of \((A^2 - B^T)C\) is:

\[
\begin{pmatrix} 162 \\ 90 \\ 420 \end{pmatrix}
\]

Let me know if you'd like more details or if you have any questions!

Here are 5 related questions to further explore:
1. How do you compute the determinant of matrix \(A\)?
2. What is the inverse of matrix \(A\) (if it exists)?
3. How do you verify the correctness of a matrix multiplication step?
4. How would the result change if \(C\) was a different vector?
5. What are the properties of matrix transposition?

**Tip**: When multiplying matrices, always check the dimensions to ensure compatibility for matrix multiplication!

Let A = ( 1 3 4 1 3 2 4 10 9 ) , B = ( 7 13 -6 -1 -7 2 -2 2 0 ) , C = ( 1 1 2 ). ii) Evaluate ( A^2 - B transpose ) C.

Learn how to solve matrix operations involving matrix multiplication, transposition, subtraction, and vector multiplication. This step-by-step solution evaluates (A^2 - B^T)C using matrix A = (1 3 4 1 3 2 4 10 9), B = (7 13 -6 -1 -7 2 -2 2 0), and vector C = (1 1 2). Suitable for advanced high school or undergraduate students.

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