Math Problem Statement

Find left parenthesis 3 right parenthesis BA plus left parenthesis 4 right parenthesis AC​, if possible. Upper A equals left bracket Start 2 By 3 Matrix 1st Row 1st Column 1 2nd Column 1 3rd Column 2 2nd Row 1st Column 0 2nd Column 0 3rd Column 2 EndMatrix right bracket

Upper B equals left bracket Start 2 By 2 Matrix 1st Row 1st Column 0 2nd Column negative 1 2nd Row 1st Column 0 2nd Column 4 EndMatrix right bracket

Upper C equals left bracket Start 3 By 3 Matrix 1st Row 1st Column negative 3 2nd Column 0 3rd Column 0 2nd Row 1st Column 3 2nd Column negative 1 3rd Column 3 3rd Row 1st Column negative 2 2nd Column 3 3rd Column 1 EndMatrix right bracket

Question content area bottom Part 1 Select the correct choice below​ and, if​ necessary, fill in the answer boxes to complete your choice. A.left parenthesis 3 right parenthesis BA plus left parenthesis 4 right parenthesis AC equals left bracket Start 2 By 3 Matrix 1st Row 1st Column negative 16 2nd Column nothing 3rd Column nothing 2nd Row 1st Column nothing 2nd Column nothing 3rd Column nothing EndMatrix right bracket left parenthesis 3 right parenthesis BA plus left parenthesis 4 right parenthesis ACequalsleft bracket Start 2 By 3 Matrix 1st Row 1st Column negative 16 2nd Column nothing 3rd Column nothing 2nd Row 1st Column nothing 2nd Column nothing 3rd Column nothing EndMatrix right bracket                  

Your answer is not correct.B. The matrix does not exist.

Solution

We are given the matrices:

A=[112002]A = \begin{bmatrix} 1 & 1 & 2 \\ 0 & 0 & 2 \end{bmatrix}

B=[0−104]B = \begin{bmatrix} 0 & -1 \\ 0 & 4 \end{bmatrix}

C=[−3003−13−231]C = \begin{bmatrix} -3 & 0 & 0 \\ 3 & -1 & 3 \\ -2 & 3 & 1 \end{bmatrix}

We need to determine if (3)BA+(4)AC(3)BA + (4)AC exists.

Step 1: Check if BABA is defined

For matrix multiplication, the number of columns in the first matrix must match the number of rows in the second matrix.

  • BB is a 2×22 \times 2 matrix.
  • AA is a 2×32 \times 3 matrix.

Since the number of columns of BB (which is 2) does not match the number of rows of AA (which is also 2), the multiplication BABA is not defined.

Step 2: Since BABA is not defined, the entire expression (3)BA+(4)AC(3)BA + (4)AC is not defined.

Thus, the correct answer is: B. The matrix does not exist.\text{B. The matrix does not exist.}

Would you like a deeper explanation of matrix multiplication rules?

Related Questions:

  1. How do you determine if two matrices can be multiplied?
  2. What are the properties of matrix multiplication?
  3. Can non-square matrices be multiplied?
  4. What happens when you multiply a matrix by a scalar?
  5. What is the importance of matrix dimensions in linear algebra?

Tip: Always check the dimensions of matrices before attempting multiplication! 🚀

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Math Problem Analysis

Mathematical Concepts

Matrix Multiplication
Scalar Multiplication
Matrix Dimensions

Formulas

Matrix multiplication condition: A (m x n) * B (n x p) = C (m x p)
Scalar multiplication: c * A

Theorems

Matrix multiplication is only defined when the number of columns of the first matrix matches the number of rows of the second matrix.

Suitable Grade Level

Grades 10-12

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