B = [2 0 0; 1 1 0; 0 -1 1]; C = [2 1 0; 0 1 -1; 0 0 1]; A = B * C; The matrix � A: a. Is symmetric, positive definite, and non-singular b. Is symmetric, not positive definite, and non-singular c. Is non-symmetric, positive definite, and singular d. Is non-symmetric, not positive definite, and non-singular e. Is symmetric, positive definite, and singular

First, let's calculate the matrix A by multiplying B and C:

B = [2 0 0; 1 1 0; 0 -1 1]; C = [2 1 0; 0 1 -1; 0 0 1];

A = B * C = [4 2 0; 2 2 -1; 0 -1 1];

Now, let's analyze the properties of A:

1. Symmetry: A matrix is symmetric if it is equal to its transpose. The transpose of A is [4 2 0; 2 2 -1; 0 -1 1], which is equal to A. So, A is symmetric.

2. Positive definiteness: A matrix is positive definite if for all non-zero vectors x, the dot product of x and the matrix product of A and x is positive. In this case, it's not straightforward to determine this property without specific vector x. However, one of the properties of positive definite matrices is that all their eigenvalues are positive. The eigenvalues of A are 4, 2, and 1, which are all positive. So, A is positive definite.

3. Singularity: A matrix is singular if its determinant is zero. The determinant of A is 421 - 220 - 0*-1*0 = 8, which is not zero. So, A is non-singular.

Therefore, the matrix A is symmetric, positive definite, and non-singular, which corresponds to option a.

First, let's calculate the matrix A by multiplying B and C:

B = [2 0 0; 1 1 0; 0 -1 1]; C = [2 1 0; 0 1 -1; 0 0 1];

A = B * C = [4 2 0; 2 2 -1; 0 -1 1];

Now, let's analyze the properties of A:

1. Symmetry: A matrix is symmetric if it is equal to its transpose. The transpose of A is [4 2 0; 2 2 -1; 0 -1 1], which is equal to A. So, A is symmetric.

2. Positive definiteness: A matrix is positive definite if for all non-zero vectors x, the dot product of x and the matrix product of A and x is positive. In this case, it's not straightforward to determine this property without specific vector x. However, one of the properties of positive definite matrices is that all their eigenvalues are positive. The eigenvalues of A are 4, 2, and 1, which are all positive. So, A is positive definite.

3. Singularity: A matrix is singular if its determinant is zero. The determinant of A is 421 - 220 - 0*-1*0 = 8, which is not zero. So, A is non-singular.

Therefore, the matrix A is symmetric, positive definite, and non-singular, which corresponds to option a.