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Chapter 9: Problem 1

Find the determinant of each matrix. $$\left[\begin{array}{ll} 1 & 3 \\ 0 & 2 \end{array}\right]$$

Short Answer

Expert verified
The determinant of the given matrix is 2.

Step by step solution

01

Identify the Matrix Dimensions

The given matrix is a 2x2 matrix. The general form of a 2x2 matrix is: \[\left[ \begin{array}{cc} a & b \ c & d \end{array} \right]\]
02

Write Down the Determinant Formula for a 2x2 Matrix

For a 2x2 matrix \( \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \), the determinant is calculated using the formula: \[ \text{det}(A) = ad - bc \]
03

Substitute the Values

Identify the elements of the matrix: \[ a = 1, b = 3, c = 0, d = 2 \] Substitute these values into the determinant formula: \[ \text{det}(A) = (1)(2) - (3)(0) \]
04

Simplify the Expression

Perform the calculations: \[ \text{det}(A) = 2 - 0 = 2 \]

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Key Concepts

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

2x2 matrix
Matrices are essential tools in algebra. A 2x2 matrix is one of the simplest forms of a matrix. It has 2 rows and 2 columns.
You'll see this type of matrix written as follows:
\(\begin{bmatrix} a & b \ c & d \end{bmatrix} \).
In this matrix:
  • 'a' represents the value in the first row, first column.
  • 'b' represents the value in the first row, second column.
  • 'c' represents the value in the second row, first column.
  • 'd' represents the value in the second row, second column.
The given matrix in the exercise is:
\(\begin{bmatrix} 1 & 3 \ 0 & 2 \end{bmatrix}\).
determinant formula
The determinant of a 2x2 matrix is a special number that can be calculated using its elements.
This determinant helps in various computations, like solving a system of linear equations.
For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated using the formula:
\( \text{det}(A) = ad - bc \).
This means you multiply 'a' and 'd,' then multiply 'b' and 'c,' and finally subtract the second product from the first.
  • 'ad' is the product of the first element in the first row and the second element in the second row.
  • 'bc' is the product of the second element in the first row and the first element in the second row.
matrix calculation
Let's apply the determinant formula to our matrix:
Given matrix: \(\begin{bmatrix} 1 & 3 \ 0 & 2 \end{bmatrix}\).
Identify the elements: \( a = 1, b = 3, c = 0, d = 2 \).
Substitute these values into the formula:
\( \text{det}(A) = (1)(2) - (3)(0) = 2 - 0 = 2 \).
So, the determinant is 2.
This calculation tells us that if we perform certain operations on rows and columns of this matrix, it maintains a specific property defined by this value.
Determinants can be useful in many areas, like finding the inverse of a matrix or solving systems of linear equations.

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