Problem 84 - College Algebra and Calculus: An Applied Approach

Question

In Exercises 83 and 84 , find a value of \(k\) that makes the matrix invertible and then find a value of \(k\) that makes the matrix singular. (There are many correct answers.) $$ \left[\begin{array}{cc} 2 k+1 & 3 \\ -7 & 1 \end{array}\right] $$

Step-by-Step Solution

Answer

The matrix is invertible for any value of \(k\) that is not equal to -11, and it's singular when \(k = -11\).

Step 1: Invertible Matrix

First, we need to calculate the determinant of the matrix \(\left[\begin{array}{cc} 2 k+1 & 3 \ -7 & 1 \end{array}\right]\). The determinant of a \(2 \times 2\) matrix \(\left[\begin{array}{cc} a & b \ c & d \end{array}\right]\) is given by \(ad - bc\). Thus, the determinant of the given matrix is \((2k+1)*1 - 3*(-7) = 2k+1 + 21 = 2k+22\). For the matrix to be invertible, the determinant must be non-zero. Therefore, for any value of \(k\) not equal to -11, the matrix will be invertible.

Step 2: Singular Matrix

For a matrix to be singular, its determinant must be zero. Using the determinant we calculated in Step 1 (2k+22), we set it equal to zero and solve for \(k\). This gives us \(2k + 22 = 0 \Rightarrow 2k = -22 \Rightarrow k = -11\). So, when \(k = -11\), the given matrix is singular.

Key Concepts

Determinant Calculation

To understand matrix algebra, knowing how to calculate the determinant is crucial. The determinant for a 2x2 matrix provides key insights into the properties of the matrix. For a matrix \[\left[\begin{array}{cc} a & b\ c & d\end{array}\right]\], the determinant \( \Delta \) can be calculated as:

• \( \Delta = ad - bc \).

The determinant is not just a number. It tells us about the matrix's behavior. It helps us find whether the matrix is invertible or singular. For example, in our given matrix:

• \( \left[\begin{array}{cc} 2k+1 & 3\ -7 & 1\end{array}\right] \)

Calculating the determinant gives:

• Determinant \( = (2k+1) \cdot 1 - 3 \cdot (-7) = 2k + 1 + 21 = 2k + 22 \).

Each value of \( k \) will change the determinant value, which in turn affects the nature of the matrix.

Singular Matrices

Singular matrices are fascinating objects in matrix algebra. A matrix is singular if it does not have an inverse. This happens when its determinant is zero. For a matrix \[\left[\begin{array}{cc} a & b\ c & d\end{array}\right]\], this means:

• The determinant \( ad - bc = 0 \).

Singular matrices have specific characteristics:

• They are associated with a loss of dimension.
• They map vectors from a space into a smaller dimension space.

With our matrix \( \left[\begin{array}{cc} 2k+1 & 3\ -7 & 1\end{array}\right] \), when \( k = -11 \), the determinant becomes zero:

• \( 2k + 22 = 0 \Rightarrow 2k = -22 \Rightarrow k = -11 \).

Hence, the matrix becomes singular, and it cannot be inverted.

Invertible Matrices

Understanding invertible matrices is vital in linear algebra. An invertible matrix, also known as a non-singular matrix, has a determinant that is not zero. This quality allows the matrix to be reversible:

• Ensure the determinant \( eq 0 \).

If a matrix \[\left[\begin{array}{cc} a & b\ c & d\end{array}\right]\] has a non-zero determinant, it is invertible:

• It means there exists another matrix that, when multiplied by it, gives the identity matrix.

For the matrix \( \left[\begin{array}{cc} 2k+1 & 3\ -7 & 1\end{array}\right] \), any \( k eq -11 \) ensures the determinant \( 2k + 22 eq 0 \), making the matrix invertible. This property allows for solving linear equations that involve such matrices, as the inverse can be used to find unknown variables.