Math Problem Statement

Find the value of x, if the matrix [[x+1, 2, 3], [4, 5, 6], [7, 8, x+9]] is a singular matrix.

Solution

The given matrix is:

x+1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & x+9 \end{pmatrix}$$ We are asked to find the value of $$x$$ such that this matrix is **singular**. A matrix is singular if and only if its determinant is zero. So, we need to calculate the determinant of the matrix and set it equal to zero to solve for $$x$$. The determinant of a $$3 \times 3$$ matrix: $$\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$$ is given by: $$\text{det} = a(ei - fh) - b(di - fg) + c(dh - eg)$$ For the given matrix: \[ \text{determinant} = (x+1)[(5)(x+9) - (6)(8)] - 2[(4)(x+9) - (6)(7)] + 3[(4)(8) - (5)(7)] \] Let's calculate each part: 1. $$(5)(x+9) - (6)(8) = 5x + 45 - 48 = 5x - 3$$ 2. $$(4)(x+9) - (6)(7) = 4x + 36 - 42 = 4x - 6$$ 3. $$(4)(8) - (5)(7) = 32 - 35 = -3$$ Now substitute these into the determinant expression: $$\text{determinant} = (x+1)(5x - 3) - 2(4x - 6) + 3(-3)$$ Expand this: $$= (x+1)(5x - 3) - 2(4x - 6) - 9$$ $$= (5x^2 - 3x + 5x - 3) - (8x - 12) - 9$$ $$= 5x^2 + 2x - 3 - 8x + 12 - 9$$ $$= 5x^2 - 6x$$ Set this equal to zero because the matrix is singular: $$5x^2 - 6x = 0$$ Factor the equation: $$x(5x - 6) = 0$$ So, $$x = 0$$ or $$x = \frac{6}{5}$$. Thus, the values of $$x$$ that make the matrix singular are $$x = 0$$ and $$x = \frac{6}{5}$$. Would you like a detailed explanation for any part? Here are 5 related questions you might explore: 1. How does the determinant change if the matrix is modified? 2. What are the geometric implications of a singular matrix? 3. How does the determinant relate to matrix invertibility? 4. Can a non-square matrix have a determinant? 5. What are some methods to compute determinants faster? **Tip:** Always check if the matrix is singular when dealing with systems of equations, as it indicates whether the system has a unique solution or not.

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Math Problem Analysis

Mathematical Concepts

Matrices
Determinants
Singular Matrices
Quadratic Equations

Formulas

Determinant of 3x3 matrix: det = a(ei - fh) - b(di - fg) + c(dh - eg)
Singular matrix: A matrix is singular if its determinant is 0

Theorems

Matrix Determinant Theorem

Suitable Grade Level

Grades 11-12