What should be the value of x so that the matrix \(\left( {\begin{array}{*{20}{c}} 2&4\\ { - 8}&{\rm{x}} \end{array}} \right)\) does not have an inverse?
-16
Finding the Value of x for a Matrix with No Inverse
A matrix does not have an inverse if and only if its determinant is zero. This condition is crucial for determining the invertibility of a matrix. We are given a 2x2 matrix:
\(\left( {\begin{array}{*{20}{c}} 2&4\\ { - 8}&{\rm{x}} \end{array}} \right)\)
For a 2x2 matrix \(\left( {\begin{array}{*{20}{c}} a&b\\ c&d \end{array}} \right)\), the determinant is calculated using the formula:
Determinant \( = ad - bc\)
In our given matrix, we have \(a = 2\), \(b = 4\), \(c = -8\), and \(d = x\). We need to calculate the determinant and set it equal to zero to find the value of x for which the matrix does not have an inverse.
Let's calculate the determinant:
Determinant \( = (2)(x) - (4)(-8)\)
Determinant \( = 2x - (-32)\)
Determinant \( = 2x + 32\)
Now, we set the determinant equal to zero because a matrix has no inverse if its determinant is zero:
\(2x + 32 = 0\)
To find the value of x, we solve this linear equation:
\(2x = -32\)
\(x = \frac{-32}{2}\)
\(x = -16\)
Therefore, when the value of x is -16, the determinant of the matrix is zero, and the matrix does not have an inverse.
Step-by-Step Solution
- Identify the given matrix and its elements. The matrix is \(\left( {\begin{array}{*{20}{c}} 2&4\\ { - 8}&{\rm{x}} \end{array}} \right)\), with \(a=2\), \(b=4\), \(c=-8\), \(d=x\).
- Recall the condition for a matrix to not have an inverse: the determinant must be zero.
- Calculate the determinant of the 2x2 matrix using the formula \(ad - bc\). Determinant \( = (2)(x) - (4)(-8) = 2x + 32\).
- Set the determinant equal to zero: \(2x + 32 = 0\).
- Solve the equation for x: \(2x = -32 \Rightarrow x = -16\).
- The value of x that makes the matrix non-invertible is -16.
Explanation of Key Concepts
- Invertible Matrix: A square matrix A is invertible (or non-singular) if there exists a matrix B such that \(AB = BA = I\), where I is the identity matrix. The matrix B is called the inverse of A, denoted \(A^{-1}\).
- Determinant: A scalar value calculated from the elements of a square matrix. It provides important information about the matrix, including whether it is invertible.
- Condition for Non-Invertibility: A square matrix is non-invertible (or singular) if and only if its determinant is zero.
Revision Table: Matrix Inverse and Determinant
| Concept | Description | Condition for 2x2 Matrix \(\left( {\begin{array}{*{20}{c}} a&b\\ c&d \end{array}} \right)\) |
|---|---|---|
| Invertible Matrix | Has an inverse matrix \(A^{-1}\). | Determinant \(\neq 0\) (\(ad-bc \neq 0\)). |
| Non-Invertible (Singular) Matrix | Does not have an inverse matrix. | Determinant \( = 0\) (\(ad-bc = 0\)). |
| Determinant of 2x2 Matrix | Scalar value calculated from matrix elements. | \(ad - bc\). |
Additional Information: Singular Matrices
A matrix that does not have an inverse is called a singular matrix. Singular matrices have a determinant of zero. They are important in linear algebra because they represent linear transformations that are not one-to-one and map vectors to a lower-dimensional space. For example, a 2x2 singular matrix maps the entire 2D plane onto a line or a point. Understanding the condition for a matrix to be singular (determinant = 0) is fundamental for solving systems of linear equations and analyzing linear transformations.
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