Problem 20 - 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. $$ \left[\begin{array}{rrr}{1} & {2} & {5} \\ {-2} & {-3} & {2} \\ {3} & {5} & {3}\end{array}\right] $$

Step-by-Step Solution

Answer

The determinant is 0, so the matrix does not have an inverse.

Step 1: Recall the Determinant Formula for a 3x3 Matrix

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

Step 2: Identify Matrix Elements

Identify and name the elements of the given matrix \( A \): \[A = \begin{bmatrix} 1 & 2 & 5 \ -2 & -3 & 2 \ 3 & 5 & 3 \end{bmatrix}.\]Here, \( a = 1 \), \( b = 2 \), \( c = 5 \), \( d = -2 \), \( e = -3 \), \( f = 2 \), \( g = 3 \), \( h = 5 \), \( i = 3 \).

Step 3: Calculate Each Component of the Determinant Formula

Solve each component of the determinant formula:1. Compute \( ei - fh: \) \[ (-3)(3) - (2)(5) = -9 - 10 = -19. \]2. Compute \( di - fg: \) \[ (-2)(3) - (2)(3) = -6 - 6 = -12. \]3. Compute \( dh - eg: \) \[ (-2)(5) - (-3)(3) = -10 + 9 = -1. \]

Step 4: Substitute Values into the Determinant Formula

Substitute the values calculated in Step 3 into the determinant formula:\[\det(A) = 1(-19) - 2(-12) + 5(-1).\]

Step 5: Simplify to Find the Determinant

Simplify the expression from Step 4:\[\det(A) = -19 + 24 - 5 = 0.\]

Step 6: Determine Invertibility from Determinant

A matrix is invertible if and only if its determinant is non-zero. Since \( \det(A) = 0 \), matrix \( A \) does not have an inverse.

Key Concepts

Invertibility

Matrix invertibility is a key concept in linear algebra. It refers to the ability of a matrix to have an inverse. An inverse matrix is like a multiplicative reciprocal for regular numbers. When you multiply a matrix by its inverse, you get the identity matrix, which is a special kind of square matrix where all diagonal elements are 1 and all other elements are 0. An important thing to remember is that only square matrices can potentially be invertible. However, even among square matrices, not all are invertible. The determinant of a matrix plays a crucial role in determining invertibility:

• If the determinant of a matrix is non-zero, the matrix is invertible.
• If the determinant is zero, the matrix is not invertible.

In our example, the determinant of the given matrix is zero. Therefore, this matrix cannot be inverted.

3x3 Matrix Determinant Formula

Determinants are handy tools in linear algebra used to determine properties of matrices, such as whether a matrix is invertible. For a 3x3 matrix, the determinant can be calculated using a specific formula. Consider a 3x3 matrix:\[A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\]To find the determinant of this matrix, use the formula:\[\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]The formula might look complex, but it's just a matter of substituting the elements of the matrix into it and performing arithmetic operations:

• Calculate \(ei - fh\)
• Calculate \(di - fg\)
• Calculate \(dh - eg\)

After substituting these calculations into the formula, you will get the determinant. For our provided matrix, this determinant evaluates to 0, signaling that the matrix lacks an inverse.

Matrix Inverse

The inverse of a matrix is a powerful concept in mathematics. It allows us to solve systems of linear equations, among many other applications. The inverse of a matrix \( A \) is denoted as \( A^{-1} \), and when \( A \) is multiplied by \( A^{-1} \), the result is the identity matrix.However, not every matrix has an inverse. As mentioned earlier, only matrices with a non-zero determinant are invertible. The process of finding an inverse for a 3x3 matrix involves several steps:

• First, determine the matrix's determinant.
• If the determinant is non-zero, perform the calculations involving the adjugate of the matrix and multiply by the reciprocal of the determinant.
• The resulting matrix will be the inverse.

For our specific case, since the determinant is zero, this matrix does not have an inverse. Thus, there is no need to perform further calculations.