Invertible Matrices

Step 1: Singularity Condition
A matrix has no inverse if its determinant is zero.
Step 2: Calculation

Step 3: Expansion

Step 4: Inspection
If we put , column 1 and column 3 become identical. , so is the real root.
Final Answer: (d)

Step 1: Understanding the condition for the inverse.
For a square matrix , the inverse exists if and only if . In this case, we are concerned with matrices where the entries of are integers, and we are analyzing the behavior of the inverse matrix based on the determinant condition.

Step 2: Matrix inversion when .
We know that the inverse of a matrix is given by: Where is the adjugate of , and its entries are polynomials of the entries of . When , this implies that the inverse matrix is simply the adjugate matrix, and all the entries in the adjugate matrix will also be integers, because the adjugate is formed from determinants of integer matrices.

Step 3: Conclusion.
Thus, if , then exists and all its entries are integers. This makes option (3) the correct answer.

Step 1: Matrix inverse property.
We are given that is the inverse of matrix , so by the property of matrix multiplication: where is the identity matrix. This means:

Step 2: Matrix multiplication.
First, we calculate the product and set it equal to the identity matrix. The matrix is given as , so we divide it by 10: Now, compute :

Step 3: Perform the multiplication.
Perform the matrix multiplication for each element of the resulting matrix: 1. First row: 2. Second row: 3. Third row:

Step 4: Set the result equal to the identity matrix.
From the matrix multiplication, we have: For , the result must be the identity matrix: Equating the elements gives us the following system of equations: After solving for

Step 4: Set the result equal to the identity matrix (contd.).
We now equate each element of the product matrix to the corresponding element of the identity matrix. This gives us the following equations: For the first row, third column: Multiplying both sides by 10: For the second row, third column: Multiplying both sides by 10: For the third row, third column: Multiplying both sides by 10:

Step 5: Conclusion.
Thus, the value of that satisfies all the equations is , corresponding to option (D).

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}

Given:


(not possible)

Step 3: Final Answer

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}





Step 3: Final Answer