Problem 15 - Algebra and Trigonometry

Question

\(15-22\) . Find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. $$ \left[\begin{array}{rrr}{2} & {1} & {0} \\ {0} & {-2} & {4} \\ {0} & {1} & {-3}\end{array}\right] $$

Step-by-Step Solution

Answer

The determinant is 4, and the matrix has an inverse.

Step 1: Identify the matrix

The given matrix is a 3x3 matrix: \[ A = \begin{bmatrix} 2 & 1 & 0 \ 0 & -2 & 4 \ 0 & 1 & -3 \end{bmatrix} \]

Step 2: Understand the determinant formula for 3x3 matrix

The determinant \( \text{det}(A) \) for a 3x3 matrix \( \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is calculated by the formula: \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Step 3: Assign values to a, b, c, d, e, f, g, h, i

From the matrix \( A \), assign the values: - \( a = 2 \), \( b = 1 \), \( c = 0 \)- \( d = 0 \), \( e = -2 \), \( f = 4 \)- \( g = 0 \), \( h = 1 \), \( i = -3 \)

Step 4: Substitute values into determinant formula

Calculate each component of the determinant formula: - \( ei - fh = (-2)(-3) - (4)(1) = 6 - 4 = 2 \)- \( di - fg = (0)(-3) - (4)(0) = 0 - 0 = 0 \)- \( dh - eg = (0)(1) - (-2)(0) = 0 - 0 = 0 \)

Step 5: Calculate the determinant

Substitute the calculated values back into the determinant formula: \[ \text{det}(A) = 2(2) - 1(0) + 0(0) = 4 \]

Step 6: Determine if the inverse exists

An inverse of a matrix exists if and only if the determinant is non-zero. Since \( \text{det}(A) = 4 \), which is not zero, the matrix \( A \) does have an inverse.

Key Concepts

Inverse of a Matrix

The inverse of a matrix is akin to finding the reciprocal of a number. For a matrix to have an inverse, it must be square, meaning it has the same number of rows and columns. Not every square matrix, however, has an inverse. For a matrix to be invertible, its determinant must not be zero.

• If the determinant is zero, the matrix is deemed "singular" or non-invertible.
• If the determinant is non-zero, then the inverse exists, although calculating it can be quite complex.

When the inverse of a matrix exists, it is represented by the original matrix raised to the power of -1, denoted as \( A^{-1} \). Unlike the straightforward reciprocal of a number, finding the inverse entails more calculations unless using advanced software tools. In practice, understanding whether an inverse exists is often more straightforward than actually computing it. This task becomes even more evident with matrices larger than 2x2.

3x3 Matrix

A 3x3 matrix is a specific type of square matrix, containing three rows and three columns. Because it is square, the possibility exists, contingent on certain conditions, that it has an inverse.
In mathematical notation, a 3x3 matrix can be written like this:

• The first row can include elements \( a, b, c \).
• The second row can include elements \( d, e, f \).
• The third row can include elements \( g, h, i \).

Hence, it looks like:\[\begin{bmatrix}a & b & c \d & e & f \g & h & i \end{bmatrix}\]The structure of this matrix implies its use in systems involving three variables in linear algebra. Such systems are common in various scientific and engineering problems. Understanding a 3x3 matrix is crucial as many real-world matrices exceed this size, and grasping this concept paves the way for tackling larger matrices.

Determinant Formula

The determinant of a matrix provides significant insights into the properties of the matrix, including whether it can be inverted. For a 3x3 matrix, the determinant is calculated using a specific formula that considers all elements of the matrix. The formula is expressed as:\[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]Here's how it works:

• First compute the minor determinants: \( ei - fh \), \( di - fg \), and \( dh - eg \).
• Then, weigh these by the elements of the first row: \( a \), \( b \), and \( c \).
• Finally, combine these to obtain the determinant value.

This formula highlights the interconnection between all elements in a 3x3 matrix and emphasizes the importance of precision in calculations. Mistakes in even a single coefficient can drastically affect the determinant and hence affect conclusions drawn about the matrix.

Non-Zero Determinant

A non-zero determinant is crucial in identifying whether a matrix has an inverse. When a matrix’s determinant does not equate to zero, it implies several important characteristics about the matrix.

• The matrix is non-singular and thus invertible.
• In terms of linear systems, this typically suggests that the system has a unique solution.
• Geometrically, it may indicate that transformations described by the matrix do not collapse space into lower dimensions.

In our given exercise, the determinant calculated was 4. Since 4 is a non-zero number, this confirms that the matrix has an inverse. So, whenever working with matrices, calculate the determinant first to determine the invertibility before attempting to find an inverse.