Matrix

Step 1: Check Statement I
A matrix is non-singular if: Inverse of a matrix is: Since , exists.

Step 2: Check Statement II
Given and are symmetric:
Step 3: Take transpose of

Step 4: Compare

Step 5: Conclusion
Since , matrix is skew-symmetric.
Final Answer:

Step 1: Definition of row matrix
A row matrix is a matrix that has exactly one row.
Step 2: General form
The order of a row matrix is:
Step 3: Analyze options

• Option (1): Only one element too restrictive ( only)
• Option (2): One row with one or more columns correct definition
• Option (3): One column with multiple rows column matrix
• Option (4): One row and one column only

Step 4: Conclusion
A row matrix has one row and one or more columns. Final Answer:

Step 1: Recall determinant properties

Step 2: Check each option
(1) Correct property. (2) Correct for matrix. (3) This is not a valid general property. (4) Correct property.
Step 3: Conclusion
Only option (3) is incorrect. Final Answer:

Step 1: Identify given orders
Given , so:
Step 2: Scalar multiplication
Multiplying a matrix by a scalar does not change its order:
Step 3: Subtraction rule
Two matrices can be subtracted only if their orders are the same, and the result has the same order.
Step 4: Resulting order
Final Answer:

Step 1: Understanding the Concept:
This question tests the formal mathematical definition of a determinant, distinguishing it from the matrix itself and specifying the conditions under which it exists.

Step 2: Key Formula or Approach:
Recall the definition: A determinant is a scalar value (a real or complex number) that can be computed from the elements of a square matrix and encodes certain properties of that matrix.

Step 3: Detailed Explanation:
Let's evaluate the validity of each option:
- (1) "Determinant is a square matrix." - This statement is incorrect. A matrix is an array or table of numbers. A determinant is a single scalar value calculated from that array. They are fundamentally different types of mathematical entities.
- (2) "Determinant is a number associated to a matrix." - While a determinant is indeed a number, this statement is too broad and technically incorrect because it implies a determinant can be associated with any matrix. Determinants are not defined for rectangular matrices (where rows columns).
- (3) "Determinant is a unique number associated to a square matrix." - This is the precise and correct definition. The determinant operation maps every square matrix to exactly one scalar value. The requirement that the matrix be square is crucial.
- (4) "Determinant is not defined for a square matrix." - This is entirely incorrect. Square matrices are the only type of matrices for which determinants are defined.

Step 4: Final Answer:
The correct statement is option (3).

Step 1: Identify definitions

• Rectangular matrix: number of rows number of columns
• Symmetric matrix:
• Scalar matrix: diagonal matrix with equal diagonal elements
• Null matrix: all elements are zero

Step 2: Match each item
(a) A matrix which is not a square matrix
(b) A square matrix
(c) Diagonal elements of a diagonal matrix are same
(d) A matrix both symmetric and skew symmetric

Step 3: Final matching
Final Answer: