If $A = \begin{bmatrix} 2 & 7 \\ 1 & 5 \end{bmatrix}$, then what is $A + 3A^{-1}$ equal to, where $A$ is a matrix of order 2?
7I
Given the matrix:
$$A = \begin{bmatrix} 2 & 7 \\ 1 & 5 \end{bmatrix}$$
Step 1: Find the determinant of \(A\)
$$|A| = (2 \times 5) - (7 \times 1) = 10 - 7 = 3$$
Step 2: Find the inverse of \(A\) (\(A^{-1}\))
For a \(2 \times 2\) matrix, \(A^{-1} = \frac{1}{|A|} \text{adj}(A)\):
$$A^{-1} = \frac{1}{3} \begin{bmatrix} 5 & -7 \\ -1 & 2 \end{bmatrix}$$
Step 3: Calculate \(3A^{-1}\)
$$3A^{-1} = 3 \times \frac{1}{3} \begin{bmatrix} 5 & -7 \\ -1 & 2 \end{bmatrix} = \begin{bmatrix} 5 & -7 \\ -1 & 2 \end{bmatrix}$$
Step 4: Calculate \(A + 3A^{-1}\)
$$A + 3A^{-1} = \begin{bmatrix} 2 & 7 \\ 1 & 5 \end{bmatrix} + \begin{bmatrix} 5 & -7 \\ -1 & 2 \end{bmatrix}$$
$$A + 3A^{-1} = \begin{bmatrix} 2+5 & 7-7 \\ 1-1 & 5+2 \end{bmatrix} = \begin{bmatrix} 7 & 0 \\ 0 & 7 \end{bmatrix}$$
Final Answer:
$$A + 3A^{-1} = 7I$$
Important Questions from Adjoint and Inverse of a Square Matrix
Let A be a matrix of order 3 × 3 and |A| = 4. If |2adj(3A)| = 2α 3β, then what is the value of (α + β)?
If A is a square matrix, then what is adj (A -1 ) – (adj A) -1 equal to?
The matrix \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} 1&3&2\\ 1&{{\rm{x}} - 1}&1\\ 2&7&{{\rm{x}} - 3} \end{array}} \right]\)
Will have inverse for every real number x except forFind the value of $|adj (2 \cdot adj A)|$ if matrix $A$ is of the order of $3$ and $|A| = 15$.
- If $A$ is a singular matrix of order $n \ge 2$, then which of the following statements is true?
