Problem 47 - Algebra and Trigonometry

Question

\(47-50=\) Solve for \(x\) $$ \left|\begin{array}{ccc}{x} & {12} & {13} \\ {0} & {x-1} & {23} \\ {0} & {0} & {x-2}\end{array}\right|=0 $$

Step-by-Step Solution

Answer

x can be 0, 1, or 2.

Step 1: Understand the Problem

We are given a 3x3 upper triangular matrix where the determinant is set equal to zero. Find the value of \(x\) that satisfies this equality.

Step 2: Use Upper Triangular Property

For an upper triangular matrix, the determinant is the product of the diagonal elements. Thus, the determinant is \(x(x-1)(x-2)\).

Step 3: Set the Determinant to Zero

Set the equation for the determinant equal to zero: \(x(x-1)(x-2) = 0\).

Step 4: Solve for x

Solve the equation \(x(x-1)(x-2) = 0\) by setting each factor equal to zero: \(x = 0\), \(x - 1 = 0\), and \(x - 2 = 0\). Thus, \(x = 0\), \(x = 1\), and \(x = 2\).

Step 5: Present the Solutions

The possible solutions for \(x\) are 0, 1, and 2.

Key Concepts

Determinant

The determinant of a matrix is a special number that can tell us a lot about the matrix. For a 3x3 matrix, the determinant can be used to understand properties such as whether the matrix is invertible. It's like a magic number. If the determinant is zero, the matrix does not have an inverse. This is why solving for determinants is so important. In our exercise, the determinant needs to be zero in order to find the values of \(x\). For upper triangular matrices, this is a straightforward task. Simply multiply the diagonal elements together and set the result to zero.

3x3 Matrix

A 3x3 matrix is a square matrix that has three rows and three columns. This results in a total of nine elements that can be used to form the matrix. In mathematics, these matrices are significant because they provide simple yet powerful ways to solve linear equations, among other tasks. Dealing with them can seem complex at first, but they follow patterns that can be learned.
For upper triangular matrices, the elements below the main diagonal are zeros, making them easier to handle. In our given exercise, the matrix is: \[ \begin{bmatrix} {x} & {12} & {13} \ {0} & {x-1} & {23} \ {0} & {0} & {x-2} \end{bmatrix} \] The structure simplifies the calculation of the determinant.

Zero Determinant

When a matrix has a determinant of zero, it implies certain properties about the matrix. One major property is that the matrix is singular, meaning it can't be inverted. This plays a crucial role in solving systems of linear equations, as it means the system doesn't have a unique solution.
In the context of our exercise, when setting the determinant to zero, it gives us specific values of \(x\) where this property holds true. The equation \(x(x-1)(x-2) = 0\) indicates that any \(x\) value that satisfies this condition leads to a zero determinant. Thus, we find \(x = 0\), \(x = 1\), and \(x = 2\) as potential solutions.

Diagonal Elements

Diagonal elements in a matrix refer to the numbers that run from the top left to the bottom right of a square matrix. They are important because, in special types of matrices like upper triangular matrices, they determine the value of the determinant.
In an upper triangular matrix, you only need to multiply these diagonal elements to find the determinant. This matrix property simplifies calculations and is especially useful for larger matrices.
In our exercise, the diagonal elements are \(x\), \(x-1\), and \(x-2\). Multiplying these gives us the determinant: \(x(x-1)(x-2)\). By setting this product to zero, we determine the possible values of \(x\) which are 0, 1, and 2.
This illustrates the key role diagonal elements play in understanding matrices, simplifying many computational aspects.