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.) $$ \left[\begin{array}{rr} 4 & 3 \\ -2 & k \end{array}\right] $$

Short answer

The matrix is invertible for any value of \(k\) unequal to -3/2. When \(k = -3/2\), the matrix becomes singular.

Step-by-step solution

- Formulate the formula for the determinant of the matrix

The determinant of a 2x2 matrix \[ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \] can be calculated with the formula \(ad-bc\). In the given matrix, \(a = 4\), \(b = 3\), \(c = -2\), and \(d = k\). This information is used to formulate the equation for the determinant of the given matrix, which is \(4k - (-2)*3 = 4k+6\).

- Solve for the invertible matrix

For the matrix to be invertible, the determinant should not be equal to 0. This requirement is expressed in this exercise by the inequality \(4k + 6 \neq 0\). Isolating the variable \(k\) yields \(k \neq -3/2\). Thus, the matrix is invertible for all \(k\) with \(k \neq -3/2\).

- Solve for the singular matrix

For the matrix to be singular, the determinant must equal zero. This requirement is expressed here by the equation \(4k + 6 = 0\). Solving for \(k\) gives \(k = -3/2\). Thus, when \(k = -3/2\), the matrix becomes singular. This result is in agreement with the finding in Step 2.

Key concepts

Determinant of a Matrix

Understanding the concept of the determinant of a matrix is crucial for anyone studying linear algebra. It is a scalar value that determines many properties of the matrix and is obtained by a specific arithmetic operation. For a 2x2 matrix, the determinant is computed using the formula \( ad - bc \) where \( a, b, c, \) and \( d \) are elements of the matrix.

The determinant carries with it significant geometric interpretation; it can represent the scaling factor for the transformation that the matrix represents, and the sign of the determinant indicates whether the transformation preserves the orientation. A nonzero determinant indicates that the matrix has an inverse, thus making it 'invertible'. Conversely, a determinant of zero implies that the matrix is 'singular' and does not have an inverse.

For instance, in our exercise, the matrix's determinant is \( 4k + 6 \) which led us to understand under what condition (if any) this particular matrix would be invertible or singular. A detailed look at the calculation of the determinant, as in the example provided, showcases how minor changes in matrix elements can lead to significant effects on its overall properties.

Singular Matrix

A singular matrix is one that does not have an inverse. One of the most fundamental properties determining a matrix's invertibility is, in fact, its determinant. A matrix is singular if and only if its determinant is zero. This is a key piece of information for systems of linear equations, as a singular matrix indicates that the system either has no solution or an infinite number of solutions.

Referring to our exercise, the singular condition is met when the determinant \( 4k + 6 \) equals zero. We find that \( k = -3/2 \) makes the matrix singular. It's essential to understand that a singular matrix cannot be used to find unique solutions in linear equations and thus identifies limitations within certain mathematical scenarios.

The concept of singularity also ties closely to matrix invertibility and matrix algebra, affecting the range of operations and transformations applicable to a given matrix. Singular matrices are a critical consideration in areas like computer graphics, physics simulations, and any field that relies on solving systems of linear equations.

Matrix Algebra

Matrix algebra is a collection of mathematical procedures involving matrices - arrays of numbers arranged in rows and columns. Operations such as addition, subtraction, and multiplication follow specific rules distinct from simple arithmetic. One of the most potent calculations in matrix algebra is finding the inverse of a matrix, which is only possible for non-singular matrices.

Key operations within matrix algebra include transposing a matrix, where the rows become columns and vice versa, as well as finding the determinant, which we explored in the context of matrix invertibility and singularity. When the determinant is nonzero, matrix algebra enables us to compute the inverse through adjugate and division by the determinant, thus allowing for a full range of linear transformations.

In our exercise, the exploration of matrix algebra concepts is exemplified by the determinant calculation and the conditions for invertibility. These algebraic manipulations form the foundation for more complex operations such as matrix factorizations, eigenvalue computation, and solving simultaneous linear equations. Mastery of matrix algebra is fundamental to areas such as engineering, data science, economics, and more.