Matrices

Step 1: Determine the dimensions of matrices A and B.
Matrix A has 3 rows and 3 columns, so its dimension is .
Matrix B has 3 rows and 2 columns, so its dimension is .

Step 2: Check the condition for each operation.
(A) For the product AB to be defined, the number of columns in A must be equal to the number of rows in B. Here, A has 3 columns and B has 3 rows, so AB is defined. The resulting matrix will have dimensions . The statement is correct.
(B) For the product BA to be defined, the number of columns in B (2) must be equal to the number of rows in A (3). Since , BA is not defined. The statement is correct.
(C) For matrix addition A + B to be defined, the matrices must have the same dimensions. A is and B is . Since their dimensions are different, A + B is not defined. The statement is correct.
(D) For matrix subtraction A - B to be defined, the matrices must have the same dimensions. Since A is and B is , A - B is not defined. The statement says that A - B is defined, which is incorrect.

Step 1: Analyze statement A.
A matrix M is skew-symmetric if , which means for all i, j. For diagonal elements, , so . This implies , so . Thus, all diagonal elements of a skew-symmetric matrix are zero. Statement A is correct.

Step 2: Analyze statement B.
A square matrix is a diagonal matrix if all of its non-diagonal elements are zero. The statement says they are one, which is incorrect. Statement B is incorrect.

Step 3: Analyze statement C.
A matrix is called singular if its determinant is zero. It is called non-singular if its determinant is non-zero. The statement claims the opposite. Statement C is incorrect.

Step 4: Analyze statement D.
A fundamental property of matrices states that for any square matrix A, , where I is the identity (unit) matrix. The statement says "unit matrix multiplied by the determinant A", which is exactly this property. Statement D is correct. Conclusion: Statements A and D are correct.