Question

Given \(A=\left[\begin{array}{cc}4 & -2 \\ 2 a-1 & 5 a-3\end{array}\right]\) and if A doesn't have multiplicative inverse, then \(12 a-13=\) (1) 6 (2) \(7 / 12\) (3) \(12 / 7\) (4) \(-6\)

Short answer

Answer: For a = \(\frac{7}{12}\), the matrix A doesn't have a multiplicative inverse, and the value of \(12a - 13\) is -6.

Step-by-step solution

Compute the determinant of A

To compute the determinant of a 2x2 matrix, we can use the formula det(A) = ad - bc, where a, b, c, and d are the elements of the matrix A. In this case, a = 4, b = -2, c = 2a - 1, and d = 5a - 3. So, det(A) = (4)(5a-3) - (-2)(2a-1) = 20a - 12 + 4a - 2 = 24a - 14.

Find the value of a for which the determinant of A is zero

We want the matrix A to not have a multiplicative inverse. For this, the determinant of A should be zero. We'll set the determinant equal to zero and solve for a: 24a - 14 = 0 Add 14 to both sides: 24a = 14 Now, divide both sides by 24: a = \(\frac{14}{24} = \frac{7}{12}\)

Compute the value of \(12a - 13\)

With the value of a found in step 2, we can compute the value of \(12a - 13\) as follows: \(12a - 13 = 12(\frac{7}{12}) - 13 = 7 - 13 = -6\) So, the value of \(12a - 13\) when A doesn't have a multiplicative inverse is -6. Therefore, the correct answer is (4) \(-6\).

Key concepts

Inverse of a Matrix

Understanding whether a matrix has an inverse hinges on a key concept in linear algebra known as the determinant. For any 2x2 matrix:\[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]The determinant is calculated using:\[det(A) = ad - bc\]A matrix possesses an inverse, if and only if its determinant is not equal to zero. If the determinant is zero, the matrix is singular, which means it doesn’t have an inverse. This property is important because in many mathematical, scientific, and engineering applications, we may need to "undo" a transformation applied by a matrix. For finding the inverse of a matrix, the determinant is the first checkpoint to ensure that the operation is possible.

Matrix Algebra

Matrix algebra is an essential part of linear algebra, dealing with operations like addition, subtraction, multiplication, and finding the determinant. These operations allow for efficient and structured handling of complex mathematical problems, simplifying systems that involve multiple variables.
- **Addition and Subtraction:** These operations are straightforward; matrices are added or subtracted element by element, which can only be performed on matrices of the same dimension. - **Multiplication:** When multiplying matrices, the elements are combined in a specific, rule-bound manner, unlike scalar multiplication. The number of columns in the first matrix must match the number of rows in the second.
Matrix multiplication is not commutative, meaning the order of multiplication matters. This property is crucial, especially when solving system of equations using matrix methods such as the inverse or determinant methods.

Solving Equations

Solving equations using matrices involves multiple techniques where matrices help simplify and organize information efficiently. One common approach is using the inverse of a matrix, particularly for linear equations in the form of:\[Ax = B\]Here, \(A\) is a known matrix, and \(x\) and \(B\) are vector or matrices. If \(A\) has an inverse (denoted as \(A^{-1}\)), then the solution \(x\) can be found as:\[x = A^{-1}B\]This approach shortens and systematizes solving process, especially for large systems. However, if the determinant of \(A\) equals zero, indicating that \(A\) doesn’t have an inverse, alternative methods like row reduction or numerical approximation methods must be used. These alternatives are important for ensuring solutions can be found even in non-standard or degenerate cases.