Matrices And Determinants

We are given that , so: Now, compute the matrix product : Thus, the correct answer is option (4).

We are given matrix: and the cofactors of entries 3, 7, and 6, which are the elements respectively. We define: We need to evaluate: Now calculate each cofactor:

So, Now compute: Final Answer:

A matrix is singular if its determinant is zero. The determinant of a 2x2 matrix is given by: We are given that , so the possible values for are all combinations of and . For to be singular, we need . This implies: Now, let’s check all the possible combinations of where : - , then , so singular. - , then , not singular. - , then , not singular. - , then , so singular. - Similarly, check other combinations. The valid combinations where are: - - These yield 8 singular matrices in total. Thus, the correct answer is option (4), 8.

We are given that is a non-singular matrix of order , and we know the property: We are asked to find . For any square matrix , the determinant of its adjugate matrix is related to the determinant of by the formula: where is the order of the matrix. Since is a matrix, , so: % Final Answer

To make matrix symmetric: So, To make skew-symmetric: Hence,

Let the given matrix be , and its inverse be . We know: Given that matrix multiplication of with its inverse equals identity matrix, one can use suitable matrix multiplication or comparison methods to find the value of . Once is found, substitute it into the 3x3 matrix: This is a known determinant form: On expansion or substitution, it simplifies to

Step 1: Understand the definition of the rank of a matrix.
The rank of a matrix is the maximum number of linearly independent row vectors or column vectors. It is also equal to the order of the largest non-zero minor (determinant of a submatrix).
The given matrix is a matrix. The maximum possible rank of this matrix is .
Step 2: Reduce the matrix to row echelon form using elementary row operations.
Let the given matrix be :
Swap and to get a '1' in the top-left corner for easier calculations: Perform row operations to make elements below the leading '1' in the first column zero:


Now, make the element in the second column of the third row zero: (to avoid fractions) Step 3: Determine the rank from the row echelon form.
The matrix is now in row echelon form. The number of non-zero rows is 3.
Therefore, the rank of the matrix is 3.
Alternative Method (using determinants of minors):
The maximum possible rank is 3. We can check if there exists a submatrix whose determinant is non-zero.
Consider the submatrix formed by the first three columns: Calculate its determinant: Since the determinant of this minor is , which is non-zero, the rank of the matrix is 3.
The final answer is .

Step 1: Recall the relationship between adjoint and determinant.
We know that: $ is the size of the matrix. Given: - , - , - . So: \) explicitly.
Compute: \) ,
.
Substitute back: \) $

We are given the cofactors of elements , in 3rd row, and of matrix as 9, 4, 3 respectively.

Use these to determine the values:

- The cofactor of is 9.
- The cofactor of is 4.
- The cofactor of is 3.

Using cofactor expansion and determinant properties, these values allow solving for .

After substitution and matrix multiplication of , we get: