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Chapter 9: Problem 16

Find the following products. $$\left[\begin{array}{ll}3 & 1 \\\4 & 0\end{array}\right]\left[\begin{array}{ll}0 & 3 \\\1 & 6\end{array}\right]$$

Short Answer

Expert verified
The product is \(\begin{bmatrix} 1 & 15 \ 0 & 12 \end{bmatrix}\).

Step by step solution

01

- Set up the matrices

Identify the two matrices to be multiplied. Matrix A is \(\begin{bmatrix} 3 & 1 \ 4 & 0 \end{bmatrix}\) and Matrix B is \(\begin{bmatrix} 0 & 3 \ 1 & 6 \end{bmatrix}\).
02

- Multiply the elements

To find the element in the first row and first column of the product matrix, calculate: \( (3 * 0) + (1 * 1) = 0 + 1 = 1 \). To find the element in the first row and second column, calculate: \( (3 * 3) + (1 * 6) = 9 + 6 = 15 \).
03

- Complete the multiplication for the second row

To find the element in the second row and first column, calculate: \( (4 * 0) + (0 * 1) = 0 + 0 = 0 \). To find the element in the second row and second column, calculate: \( (4 * 3) + (0 * 6) = 12 + 0 = 12 \).
04

- Combine the results

Combine all calculated elements into the resulting matrix: \( \begin{bmatrix} 1 & 15 \ 0 & 12 \end{bmatrix} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

linear algebra
Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations.

Matrices play a significant role in it. Matrices are used to represent and manipulate linear equations and transformations.
  • For example, they can represent systems of linear equations, transformations in geometry, and even complex networks.
Understanding matrix multiplication, element-wise multiplication, and other operations is essential when studying linear algebra.

Here are some key points to remember:
  • Matrix product involves rows and columns.
  • Element-wise multiplication is done element by element.
  • Linear transformations often use matrices as representation tools.
Building a solid foundation in linear algebra will help in many practical applications, from computer graphics to machine learning.

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Most popular questions from this chapter

Solve each system, using Cramer's rule where possible. $$\begin{aligned} x+y-2 z &=1 \\ x-2 y+z &=2 \\ 2 x-y-z &=3 \end{aligned}$$

Solve each system of equations by using } A^{-1} \text { if possible.}$$ $$\begin{array}{l} x+y-z=4 \\ 2 x-3 y+z=2 \\ 4 x-y-z=6 \end{array}$$

The equation of a line through two points can be expressed as an equation involving a determinant. Show that the following equation is equivalent to the equation of the line through \((3,-5)\) and \((-2,6)\). $$\left|\begin{array}{rrr} x & y & 1 \\ 3 & -5 & 1 \\ -2 & 6 & 1 \end{array}\right|=0$$

$$\begin{aligned} &0.5 x-0.25 y+0.1 z=3\\\ &0.2 x-0.5 y+0.2 z=-2\\\ &0.1 x+0.3 y-0.5 z=-8 \end{aligned}$$

Find each product if possible. $$\left[\begin{array}{r}-5 \\\1 \\\4\end{array}\right]\left[\begin{array}{lll}-1 & 0 & 3\end{array}\right]$$
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