Solving Determinant Equations And Matrix Operations A Comprehensive Guide

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This article delves into the solutions of a determinant equation and explores various matrix operations. We will address two key questions, providing a comprehensive, SEO-optimized explanation suitable for students and enthusiasts alike. This article aims to dissect these mathematical concepts, providing step-by-step solutions and insights into the underlying principles.

H2: Question 1 (a) Solving for x in a Determinant Equation

Determinant equations can seem daunting at first, but with a systematic approach, they become manageable. In this section, we will tackle the problem of solving for x given the following determinant equation:

| x+1   3   5 |
| 2   x+2   5 |
| 2   3   x+4 |
``` = 0

To solve for *x*, we need to expand this 3x3 determinant and set the resulting expression equal to zero. The expansion of a 3x3 determinant involves multiplying elements along diagonals and applying alternating signs. This process can be broken down into several steps to ensure accuracy and clarity.

### H3: Step-by-Step Expansion of the Determinant

First, let's expand the determinant along the first row. This involves multiplying each element in the first row by its corresponding minor (the determinant of the 2x2 matrix obtained by removing the row and column of that element) and applying the correct sign. The expansion is as follows:

(x+1) * | x+2  5 | - 3 * | 2  5 | + 5 * | 2  x+2 |

| 3 x+4 |       | 2 x+4 |     | 2 3 |

Now, let's compute the 2x2 determinants:

*   |(x+2)(x+4) - 5*3| = (x+2)(x+4) - 15 = x^2 + 6x + 8 - 15 = x^2 + 6x - 7
*   |(2(x+4) - 5*2| = 2x + 8 - 10 = 2x - 2
*   |(2*3 - 2(x+2)| = 6 - 2x - 4 = 2 - 2x

Substituting these back into the expansion, we get:

(x+1)(x^2 + 6x - 7) - 3(2x - 2) + 5(2 - 2x) = 0

### H3: Simplifying the Equation

Now, let's expand and simplify the equation:

(x^3 + 6x^2 - 7x + x^2 + 6x - 7) - 6x + 6 + 10 - 10x = 0
x^3 + 7x^2 - x - 7 - 6x + 6 + 10 - 10x = 0
x^3 + 7x^2 - 17x + 9 = 0

This results in a cubic equation. Solving cubic equations can be complex, but we can often find integer roots by using the Rational Root Theorem or by testing factors of the constant term (9 in this case). Let's test some factors:

*   If x = 1: 1 + 7 - 17 + 9 = 0. So, x = 1 is a root.

Since x = 1 is a root, (x - 1) is a factor of the cubic equation. We can perform polynomial division to find the other factors:

(x^3 + 7x^2 - 17x + 9) / (x - 1) = x^2 + 8x - 9

Now, we need to solve the quadratic equation x^2 + 8x - 9 = 0. This can be factored as:

(x + 9)(x - 1) = 0

So, the roots of the quadratic equation are x = -9 and x = 1. Thus, the solutions for the original cubic equation are x = 1, x = 1, and x = -9.

### H3: Final Solutions for x

Therefore, the solutions for *x* in the given determinant equation are **x = 1 (with multiplicity 2) and x = -9**. This detailed step-by-step solution ensures that the process is clear and easy to follow, making it an excellent resource for anyone studying determinants and algebra.

## H2: Question 1 (b) Matrix Operations

In this section, we delve into matrix operations. Given the matrix:

A = | 11 10 8 | | 6 -16 -1 | | 7 1 5 |


We will explore various operations that can be performed on this matrix. Understanding matrix operations is crucial in various fields, including computer graphics, engineering, and economics. This section will cover the fundamental operations and their applications.

### H3: Basic Matrix Operations

Matrix operations include addition, subtraction, scalar multiplication, and matrix multiplication. Let's briefly discuss each of these operations:

1.  **Addition and Subtraction**: To add or subtract matrices, they must have the same dimensions. The corresponding elements are added or subtracted.
2.  **Scalar Multiplication**: This involves multiplying each element of the matrix by a scalar (a constant). For example, multiplying matrix A by a scalar *k* results in a new matrix where each element *a_ij* is replaced by *k* *a_ij*.
3.  **Matrix Multiplication**: This is a more complex operation. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. The resulting matrix has dimensions equal to the number of rows in the first matrix and the number of columns in the second matrix.

### H3: Further Operations and Properties

Beyond the basic operations, matrices have several important properties and operations:

*   **Transpose**: The transpose of a matrix is obtained by interchanging its rows and columns. The transpose of matrix A, denoted as A^T, is a matrix where the *i*-th row becomes the *i*-th column of A^T.
*   **Determinant**: The determinant of a square matrix is a scalar value that can be computed using various methods. For a 3x3 matrix, the determinant can be calculated as discussed in Question 1(a). The determinant provides valuable information about the matrix, such as whether it is invertible.
*   **Inverse**: The inverse of a square matrix A, denoted as A^-1, is a matrix such that A * A^-1 = A^-1 * A = I, where I is the identity matrix. A matrix has an inverse if and only if its determinant is non-zero.
*   **Eigenvalues and Eigenvectors**: These are fundamental concepts in linear algebra. Eigenvalues are scalars, and eigenvectors are non-zero vectors that, when multiplied by the matrix, result in a scaled version of the same vector. Eigenvalues and eigenvectors are used in various applications, including stability analysis and principal component analysis.

### H3: Example: Scalar Multiplication and Transpose

To illustrate these operations, let’s consider scalar multiplication and transpose with matrix A.

1.  **Scalar Multiplication**: Let's multiply matrix A by the scalar 2:

2A = 2 * | 11 10 8 | = | 22 20 16 | | 6 -16 -1 | | 12 -32 -2 | | 7 1 5 | | 14 2 10 |


2.  **Transpose**: The transpose of matrix A is:

A^T = | 11 6 7 | | 10 -16 1 | | 8 -1 5 |


These examples demonstrate how simple scalar multiplication and transposition can be performed on a matrix. Understanding these operations is essential for more advanced matrix manipulations.

### H3: Importance of Matrix Operations

Matrix operations are fundamental tools in various applications. They are used in solving systems of linear equations, representing transformations in computer graphics, and analyzing data in statistics. A solid understanding of matrix operations is essential for anyone working in these fields.

## H2: Conclusion

In conclusion, this article has provided detailed solutions and explanations for solving a determinant equation and performing matrix operations. We tackled the problem of finding *x* in a 3x3 determinant equation and explored various matrix operations, including scalar multiplication and transposition. These concepts are crucial for a strong foundation in mathematics and its applications. By breaking down complex problems into manageable steps, this guide aims to enhance understanding and proficiency in these areas.

This comprehensive explanation, optimized for SEO, serves as a valuable resource for students and anyone interested in delving deeper into the world of mathematics and matrix operations.