Problem 71 - College Algebra

Question

Which one of the following is true? a. Some nonsquare matrices have inverses. b. All square \(2 \times 2\) matrices have inverses because there is a formula for finding these inverses. \- c. Two \(2 \times 2\) invertible matrices can have a matrix sum that is not invertible. -d. To solve the matrix equation \(A X=B\) for \(X,\) multiply \(A\) and the inverse of \(B\)

Step-by-Step Solution

Answer

Only the third statement is true. Two \(2 \times 2\) invertible matrices can have a matrix sum that is not invertible.

Step 1: Check truth of first statement

The first statement is that some nonsquare matrices have inverses. By definition, an inverse of a matrix exists only if the matrix is square – that is, has equal numbers of rows and columns. Therefore, nonsquare matrices do not have inverses, making this statement false.

Step 2: Check truth of second statement

The second statement is that all square \(2 \times 2\) matrices have inverses because there is a formula for finding these inverses. This statement is not universally true. While there is indeed a formula for finding the inverse of a \(2 \times 2\) matrix, not all \(2 \times 2\) matrices have an inverse. Specifically, a matrix has an inverse only if its determinant is nonzero. Hence, this statement is false.

Step 3: Check truth of third statement

The third statement is that two \(2 \times 2\) invertible matrices can have a matrix sum that is not invertible. This statement is potentially true: it is possible for the sum of two invertible (i.e., nonsingular) matrices to be noninvertible. An example would be if we sum an identity matrix \(I\) and \(-I\), which gives the zero matrix which is non-invertible. Therefore, this statement is true.

Step 4: Check truth of fourth statement

The fourth statement is that to solve the matrix equation \(AX = B\) for \(X\), we should multiply \(A\) and the inverse of \(B\). This statement is false. To solve the matrix equation \(AX = B\) for \(X\), we multiply both sides of the equation by the inverse of \(A\). The correct operation is \(X = A^{-1} B\), not a multiplication of \(A\) and \(B^{-1}\)

Key Concepts

Matrix Algebra

Matrix algebra forms the cornerstone of linear algebra and is crucial in a multitude of disciplines such as physics, economics, engineering, and more.

In essence, matrix algebra deals with the algebra of matrices, meaning the various operations that can be performed on matrices, including addition, subtraction, multiplication, and finding inverses. Matrices hold a particular set of elements arranged in rows and columns, and each operation follows specific rules.

For instance, to add or subtract matrices, they must have the same dimensions. Similarly, multiplication requires the number of columns in the first matrix to match the number of rows in the second. A crucial aspect is that matrix multiplication is not commutative, meaning that the order of multiplication matters: generally, \(AB eq BA\).

Understanding these operations is essential for solving systems of linear equations, transforming geometric figures, and modeling real-world problems using linear expressions.

Determinant of a Matrix

The determinant is a scalar attribute of a square matrix that provides important properties of the matrix and is signified by \(det(A)\) or \(|A|\). It holds a key role in matrix theory and has practical applications in analytical geometry, differential equations, and more.

To grasp the concept of an inverse of a matrix, understanding the determinant is fundamental. For a \(2 \times 2\) matrix, the determinant is calculated as \(ad - bc\) given that the matrix is in the form:\[\begin{pmatrix} a & b \ c & d \end{pmatrix}\].

If the determinant is zero, the matrix is termed 'singular', meaning it does not have an inverse. A non-zero determinant indicates the matrix is 'nonsingular' and hence invertible. The determinant can also give us insight into the geometrical properties of the matrix, such as scaling factors and the orientation of the transformed space.

Invertible Matrices

Invertible matrices, or nonsingular matrices, are square matrices that possess an inverse. The inverse of a matrix \(A\) is a special matrix, denoted as \(A^{-1}\), that when multiplied with \(A\) yields the identity matrix \(I\), the matrix equivalent of the number '1' in arithmetic.

Finding the inverse of a matrix is valuable for solving equations of the form \(AX = B\). Here, we can find \(X\), the unknown, by multiplying both sides of the equation by \(A^{-1}\), thus obtaining \(X = A^{-1}B\). The inverse of a matrix is only possible to find if the determinant is non-zero. This ties the concepts of determinant and invertibility closely together.

However, it's important to note that not all operations with invertible matrices result in an invertible matrix. For instance, even if two matrices are invertible on their own, their sum may not necessarily be invertible – as can be seen with the identity matrix \(I\) and its negative \(-I\), which sum to the zero matrix that lacks an inverse.