Calculating The Determinant Of 3A + 5I A Step-by-Step Guide

In linear algebra, matrices are fundamental mathematical objects that are used to represent linear transformations and systems of linear equations. The determinant of a matrix is a scalar value that provides important information about the matrix, such as whether the matrix is invertible and the volume scaling factor of the linear transformation represented by the matrix. This article delves into the process of calculating the determinant of a matrix expression, specifically focusing on the expression |3A + 5I|, where A is a given matrix and I is the identity matrix.

Problem Statement

Given the matrix:

$A = \begin{pmatrix} 2 & 3 \\ 0 & 4 \\ \end{pmatrix}$

We aim to find the value of |3A + 5I|, where I is the 2x2 identity matrix.

Prerequisites

Before diving into the solution, it's essential to understand the following concepts:

1. Matrix Multiplication: The product of a scalar and a matrix is obtained by multiplying each element of the matrix by the scalar.

2. Matrix Addition: The sum of two matrices of the same dimensions is obtained by adding corresponding elements.

3. Identity Matrix: The identity matrix, denoted by I, is a square matrix with ones on the main diagonal and zeros elsewhere. For a 2x2 matrix, the identity matrix is:

   $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix}$

4. Determinant of a 2x2 Matrix: The determinant of a 2x2 matrix $M = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}$ is given by |M| = ad - bc.

Step-by-Step Solution

1. Calculate 3A

To find 3A, we multiply each element of matrix A by the scalar 3:

$3A = 3 \times \begin{pmatrix} 2 & 3 \\ 0 & 4 \\ \end{pmatrix} = \begin{pmatrix} 3 \times 2 & 3 \times 3 \\ 3 \times 0 & 3 \times 4 \\ \end{pmatrix} = \begin{pmatrix} 6 & 9 \\ 0 & 12 \\ \end{pmatrix}$

2. Calculate 5I

Next, we find 5I by multiplying each element of the 2x2 identity matrix by the scalar 5:

$5I = 5 \times \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix} = \begin{pmatrix} 5 \times 1 & 5 \times 0 \\ 5 \times 0 & 5 \times 1 \\ \end{pmatrix} = \begin{pmatrix} 5 & 0 \\ 0 & 5 \\ \end{pmatrix}$

3. Calculate 3A + 5I

Now, we add the matrices 3A and 5I:

$3A + 5I = \begin{pmatrix} 6 & 9 \\ 0 & 12 \\ \end{pmatrix} + \begin{pmatrix} 5 & 0 \\ 0 & 5 \\ \end{pmatrix} = \begin{pmatrix} 6+5 & 9+0 \\ 0+0 & 12+5 \\ \end{pmatrix} = \begin{pmatrix} 11 & 9 \\ 0 & 17 \\ \end{pmatrix}$

4. Calculate |3A + 5I|

Finally, we compute the determinant of the resulting matrix:

$|3A + 5I| = \begin{vmatrix} 11 & 9 \\ 0 & 17 \\ \end{vmatrix} = (11 \times 17) - (9 \times 0) = 187 - 0 = 187$

Therefore, the value of |3A + 5I| is 187.

Alternative Approach: Properties of Determinants

While the direct calculation method is straightforward, understanding the properties of determinants can offer alternative approaches and insights. However, in this specific case, applying determinant properties might not simplify the calculation significantly, but it's worth mentioning for a comprehensive understanding.

One relevant property is that for a scalar k and an n x n matrix A:

$|kA| = k^n |A|$

However, this property doesn't directly apply to |3A + 5I| because of the addition operation. Determinant properties are more readily applicable to products and scalar multiples of matrices, not sums.

Common Mistakes and How to Avoid Them

1. Incorrect Scalar Multiplication: Ensure that each element of the matrix is multiplied by the scalar.
2. Incorrect Matrix Addition: Ensure that you are adding corresponding elements of the matrices.
3. Incorrect Determinant Calculation: Double-check the formula for the determinant (ad - bc) and the arithmetic.
4. Misapplication of Determinant Properties: Be cautious when applying determinant properties. They are not universally applicable to all matrix operations, especially addition.

Importance of Determinants

The determinant of a matrix is a crucial concept in linear algebra with numerous applications:

1. Invertibility: A square matrix is invertible if and only if its determinant is non-zero. This is a fundamental criterion for solving systems of linear equations.
2. Volume Scaling: The absolute value of the determinant represents the factor by which the linear transformation associated with the matrix scales volumes. If the determinant is zero, the transformation collapses space, reducing dimensionality.
3. Eigenvalues: Determinants play a key role in finding eigenvalues of a matrix, which are essential in various applications, including stability analysis of systems.
4. Change of Variables: In multivariable calculus, determinants appear in the Jacobian matrix, which is used for changing variables in multiple integrals.

Applications in Real-World Scenarios

1. Computer Graphics: Determinants are used to determine the orientation of polygons and to perform transformations in 2D and 3D graphics.
2. Physics: In physics, determinants are used in mechanics, electromagnetism, and quantum mechanics to solve various problems involving linear transformations and systems.
3. Engineering: Engineers use determinants in structural analysis, control systems, and circuit analysis.
4. Economics: Economists use determinants in input-output analysis and game theory.

Practice Problems

To reinforce your understanding, try solving the following problems:

1. Given $B = \begin{pmatrix} 1 & -2 \\ 3 & 4 \\ \end{pmatrix}$, find |2B + 3I|.
2. Given $C = \begin{pmatrix} 5 & 2 \\ -1 & 3 \\ \end{pmatrix}$, find |4C - 2I|.
3. If $D = \begin{pmatrix} -2 & 1 \\ 0 & 5 \\ \end{pmatrix}$, calculate |3D + 4I|.

Conclusion

Calculating the determinant of a matrix expression like |3A + 5I| involves scalar multiplication, matrix addition, and the application of the determinant formula. This process highlights the fundamental operations in linear algebra and the significance of determinants in various mathematical and real-world contexts. By understanding these concepts and practicing problem-solving, one can gain a deeper appreciation for the power and versatility of matrices and determinants. The determinant, as a scalar value derived from a square matrix, encapsulates essential properties of the matrix and its associated linear transformation, making it a cornerstone of linear algebra and its applications.

Understanding how to manipulate matrices and calculate their determinants is not just an academic exercise; it's a crucial skill for anyone working in fields that rely on mathematical modeling and computation. The ability to break down complex problems into matrix operations and interpret the results is a valuable asset in many scientific and technical disciplines. Furthermore, the determinant's role in determining invertibility and volume scaling provides a geometrical intuition that enriches one's understanding of linear transformations.

In summary, the calculation of |3A + 5I| serves as a microcosm of the broader world of linear algebra, illustrating the interplay between algebraic manipulations and geometric interpretations. By mastering these fundamental concepts, one can unlock a powerful toolkit for solving a wide range of problems across various domains.