Question

Find the determinant of the \(2 \times 2\) matrix. \(\left[\begin{array}{cc}-10 & -1 \\ -4 & 7\end{array}\right]\)

Short answer

The determinant of the matrix is \(-74\).

Step-by-step solution

Identify the Formula for Determinant of a 2x2 Matrix

To find the determinant of a 2x2 matrix \( \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \), use the formula: \( \text{det} = ad - bc \). This formula is specific for 2x2 matrices.

Assign Values from the Matrix to Variables

Given the matrix \( \left[ \begin{array}{cc} -10 & -1 \ -4 & 7 \end{array} \right] \), assign: \( a = -10 \), \( b = -1 \), \( c = -4 \), and \( d = 7 \).

Substitute Values into the Determinant Formula

Substitute the values into the formula: \( \text{det} = ad - bc = (-10)(7) - (-1)(-4) \).

Calculate the Products

Calculate the products: \((-10)(7) = -70\) and \((-1)(-4) = 4\).

Compute the Determinant

Subtract the second product from the first: \( -70 - 4 = -74 \). Thus, the determinant of the matrix is \(-74\).

Key concepts

2x2 matrix

A **2x2 matrix** is one of the simplest types of matrices in matrix algebra. It is a rectangular array of numbers that has two rows and two columns.
Each element in the matrix is typically a number or sometimes a symbolic representation.

• In a 2x2 matrix, you have four elements in total, arranged in a form like this: \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]
• The position of each number is important, as it is used in computations involving the matrix.
• The rows run horizontally, and the columns run vertically. This means that in our example above, 'a' and 'b' are in the first row, and 'c' and 'd' are in the second row.
• Similarly, 'a' and 'c' are in the first column, while 'b' and 'd' belong to the second column.

This simple structure makes the 2x2 matrix a perfect starting point for learning more about matrices and their applications.

determinant formula

The **determinant formula** for a 2x2 matrix is a specific rule used to find the determinant, which is a special number that can be calculated from the elements of a square matrix. For a matrix like this:
\[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]
the determinant, often represented as \( \text{det} \), can be calculated using the simple formula: \( \text{det} = ad - bc \).
Here is how it works:

• Multiply the top left element \(a\) by the bottom right element \(d\).
• Multiply the top right element \(b\) by the bottom left element \(c\).
• Subtract the second product from the first.

This gives you the determinant of the matrix. This formula is particularly easy to remember and apply, making it a valuable tool in elementary matrix algebra.

matrix algebra

**Matrix algebra** involves a set of mathematical operations that can be performed on matrices, including addition, subtraction, multiplication, and finding determinants. These operations are foundational in various fields such as physics, computer science, and economics.

• In matrix addition and subtraction, corresponding elements are simply added or subtracted.
• Matrix multiplication is a bit more complex and involves summing the products of corresponding elements from rows and columns.
• Finding the determinant is another operation that assesses a square matrix. It provides useful information about the matrix, like if it's invertible or the volume factor associated with the linear transformation the matrix represents.

In the case of a 2x2 matrix, matrix algebra is fairly straightforward. The operations are simple enough to be performed by hand but understanding them is crucial for dealing with larger matrices or more complex applications. Mastery of matrix algebra opens up many possibilities for solving real-world problems efficiently.