The Given Determinant Equation

Oct 13, 2025 by talynt.org 13 views

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Introduction

Determinants are a crucial concept in linear algebra, and solving determinant equations is an essential skill for any math enthusiast. In this article, we will delve into the world of determinants and explore a step-by-step guide on how to solve the given determinant equation.

What is a Determinant?

A determinant is a scalar value that can be calculated from the elements of a square matrix. It is denoted by the symbol |A| or det(A). The determinant of a 3x3 matrix A can be calculated using the formula:

|A| = a(ei − fh) − b(di − fg) + c(dh − eg)

where a, b, c, d, e, f, g, h, and i are the elements of the matrix.

The Given Determinant Equation

The given determinant equation is:

| ${\left|\begin{array}{ccc} x+1 & 6 & 2 \\ 1-x & -5 & x-1 \\ x-1 & 4 & 0 \end{array}\right|=0}$

Our goal is to find the values of x that satisfy this equation.

Expanding the Determinant

To solve the determinant equation, we need to expand the determinant using the formula:

|A| = a(ei − fh) − b(di − fg) + c(dh − eg)

Applying this formula to the given determinant, we get:

| [\left|\begin{array}{ccc} x+1 & 6 & 2 \ 1-x & -5 & x-1 \ x-1 & 4 & 0 \end{array}\right| = (x+1)((-5)(0) - (x-1)(4)) - 6((1-x)(0) - (x-1)(x-1)) + 2((1-x)(4) - (-5)(x-1))

Simplifying the Expression

Expanding and simplifying the expression, we get:

(x+1)(-4x+4) - 6(0 - (x-1)^2) + 2(4 - 5x + 5)

Combine like terms and simplify further:

-4x^2 + 4x - 4x + 4 - 6(-x^2 + 2x - 1) + 8 - 10x + 10

Combine like terms and simplify further:

-4x^2 + 4x - 4x + 4 + 6x^2 - 12x + 6 + 8 - 10x + 10

Combine like terms and simplify further:

2x^2 - 16x + 28

Setting the Expression Equal to Zero

Now that we have simplified the expression, we can set it equal to zero and solve for x:

2x^2 - 16x + 28 = 0

Solving the Quadratic Equation

To solve the quadratic equation, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 2, b = -16, and c = 28. Plugging these values into the formula, we get:

x = (16 ± √((-16)^2 - 4(2)(28))) / 2(2)

x = (16 ± √(256 - 224)) / 4

x = (16 ± √32) / 4

x = (16 ± 4√2) / 4

x = 4 ± √2

Conclusion

In this article, we have solved the determinant equation using a step-by-step guide. We expanded the determinant, simplified the expression, and set it equal to zero. We then solved the resulting quadratic equation using the quadratic formula. The solutions to the equation are x = 4 + √2 and x = 4 - √2.

Final Answer

The final answer is x = 4 + √2 and x = 4 - √2.

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Introduction

In our previous article, we solved the determinant equation using a step-by-step guide. However, we understand that some readers may still have questions about the topic. In this article, we will address some of the most frequently asked questions about determinant equations.

Q: What is a determinant?

A: A determinant is a scalar value that can be calculated from the elements of a square matrix. It is denoted by the symbol |A| or det(A).

Q: How do I calculate the determinant of a 3x3 matrix?

A: To calculate the determinant of a 3x3 matrix, you can use the formula:

|A| = a(ei − fh) − b(di − fg) + c(dh − eg)

where a, b, c, d, e, f, g, h, and i are the elements of the matrix.

Q: What is the difference between a determinant and a matrix?

A: A matrix is a rectangular array of numbers, while a determinant is a scalar value that can be calculated from the elements of a matrix.

Q: How do I expand a determinant?

A: To expand a determinant, you can use the formula:

|A| = a(ei − fh) − b(di − fg) + c(dh − eg)

where a, b, c, d, e, f, g, h, and i are the elements of the matrix.

Q: What is the purpose of expanding a determinant?

A: Expanding a determinant allows you to simplify the expression and solve for the unknown variables.

Q: How do I simplify a determinant expression?

A: To simplify a determinant expression, you can combine like terms and use algebraic manipulations to reduce the expression to its simplest form.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that allows you to solve quadratic equations of the form ax^2 + bx + c = 0.

Q: How do I use the quadratic formula to solve a determinant equation?

A: To use the quadratic formula to solve a determinant equation, you can first expand the determinant and simplify the expression. Then, you can set the expression equal to zero and use the quadratic formula to solve for the unknown variables.

Q: What are the solutions to the determinant equation?

A: The solutions to the determinant equation are x = 4 + √2 and x = 4 - √2.

Q: How do I verify the solutions to the determinant equation?

A: To verify the solutions to the determinant equation, you can plug the values of x back into the original equation and check if the equation holds true.

Conclusion

In this article, we have addressed some of the most frequently asked questions about determinant equations. We hope that this article has provided you with a better understanding of the topic and has helped you to solve your own determinant equations.

Final Answer

The final answer is x = 4 + √2 and x = 4 - √2.