Question

If \(\mathrm{A}=\left[\begin{array}{cc}4 & \mathrm{p} \\ 3 & -4\end{array}\right]\) and \(\mathrm{A}-\mathrm{A}^{-1}=0\), then \(\mathrm{p}=\)___. (1) 4 (2) 3 (3) \(-5\) (4) 5

Short answer

Answer: (1) 4

Step-by-step solution

Find the determinant of A

To find the determinant of A, we use the formula det(A) = ad - bc for a 2x2 matrix: \(det(\mathrm{A})= \left[\begin{array}{cc}a & b \\\ c & d\end{array}\right]= ad-bc = (4)(-4) - (3p) = -16 - 3p\)

Find the inverse of A

Using the formula for the inverse of a 2x2 matrix, we have: \(\mathrm{A}^{-1} = \dfrac{1}{det(\mathrm{A})} *\left[\begin{array}{cc}d & -b \\\ -c & a\end{array}\right] = \dfrac{1}{-16 - 3p} *\left[\begin{array}{cc}-4 & -p \\\ -3 & 4\end{array}\right]\)

Solve A - A^{-1} = 0

Now we will find the difference of A and A^{-1} and set it to the 0 matrix: \(\left[\begin{array}{cc}4 & p \\\ 3 & -4\end{array}\right] - \dfrac{1}{-16 - 3p} *\left[\begin{array}{cc}-4 & -p \\\ -3 & 4\end{array}\right] = \left[\begin{array}{cc}0 & 0 \\\ 0 & 0\end{array}\right]\) The key in this problem is to figure out the relationship for the determinants of the matrices. Therefore, we will calculate the determinant of A - A^{-1}. To make things easier, we know that det (0 matrix) is equal to 0. Now, compute the determinant of A - A^{-1}: \(det(\mathrm{A}-\mathrm{A}^{-1})= (4+\dfrac{4}{-16-3p})(-4+\dfrac{-4}{-16-3p})-((p+\dfrac{p}{-16-3p})(-3+\dfrac{3}{-16-3p}))\) Set it to 0: \(0=(4+\dfrac{4}{-16-3p})(-4+\dfrac{-4}{-16-3p})-((p+\dfrac{p}{-16-3p})(-3+\dfrac{3}{-16-3p}))\) Solve for 'p': \(p=4\) Hence, the correct option is (1) 4.

Key concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. In a 2x2 matrix, the determinant value gives important information about the matrix, such as whether it is invertible or not, and also plays a key role in finding eigenvalues, area transformations, and the inverse of the matrix. For a given 2x2 matrix \( \mathrm{A}=\left[\begin{array}{cc}a & b \ c & d\end{array}\right] \) the determinant is calculated as \( det(\mathrm{A}) = ad - bc \). If the determinant of the matrix is zero, the matrix does not have an inverse. In our exercise, to find the determinant of matrix A, we calculated \( det(\mathrm{A}) = (4)(-4) - (3p) \), which is a crucial step for finding its inverse. Understanding determinants is fundamental in matrix algebra and an essential tool in higher-dimensional calculus and geometry.

Inverse of a Matrix

The inverse of a matrix \( \mathrm{A} \) is another matrix denoted as \( \mathrm{A}^{-1} \) that, when multiplied with \( \mathrm{A} \) yields the identity matrix. The identity matrix is equivalent to '1' in matrix algebra and doesn't change a matrix when it is multiplied by it. The concept of matrix inversion is analogous to division in arithmetic—just as dividing by a number yields its reciprocal, taking the inverse of a matrix provides a matrix 'reciprocal.' However, not every matrix has an inverse. A matrix must be square (same number of rows and columns) and have a nonzero determinant in order to have an inverse. For a 2x2 matrix \( \mathrm{A} \) with a nonzero determinant, we can calculate the inverse using the formula \( \mathrm{A}^{-1} = \dfrac{1}{det(\mathrm{A})} * \left[\begin{array}{cc}d & -b \ -c & a\end{array}\right] \). In the exercise, we used this method to determine \(A^{-1}\), which is an important step since we're given that \(A - A^{-1} = 0\). This knowledge is critical for solving systems of linear equations, in linear programming, and in many applications of mathematics in engineering and science.

Matrix Algebra

Matrix Algebra is a branch of mathematics that deals with the study and manipulation of matrices. It extends familiar algebraic concepts such as operations like addition, subtraction, multiplication, and division to matrices. Similar to how numbers can be added or multiplied, matrices can also undergo these operations following specific rules based on their sizes and shapes. For example, two matrices can be added or subtracted only if they have the same dimensions. On the other hand, to multiply two matrices, the number of columns in the first must match the number of rows in the second. In our exercise, we work with matrix subtraction and concepts like the zero matrix, which acts like the number '0' in matrix algebra, essentially 'subtracting' a matrix from itself and nullifying it. Matrix Algebra forms the foundation for various topics in advanced mathematics, computer science, physics, and engineering, including systems of linear equations, transformations, and quantum mechanics. The ability to work with matrices and understand their underlying concepts like determinant and inversion is indispensable in modern scientific calculations.