1)The dimensions of ST.
S[7/12 -1/9]
[-1/3 1/9]
T[4 -5 2]
[8 -1 3]
answer=2x3
2)The pair of matrices that are inverses.
P[3 1] Q[4 4] R[0 -1/4]
[-4 0] [12 21] [1 3/4]
S[7/12 -1/9] T[4 -5 2] U[-9 6 4]
[-1/3 1/9] [8 -1 3] [-5 -2 3]
V[3 1]
[0 2]
[-4 5]
answer=S and R
3)Find the value of [15 4]
[-6 -2]
answer=-6
4)Evaluate:[1 4 -1]
[0 3 5]
answer=10
1) To find the dimensions of the matrix ST, we count the number of rows and columns. The matrix S has 2 rows and 2 columns, and the matrix T has 2 rows and 3 columns. To multiply these matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. So, in this case, we can multiply ST. The resulting matrix will have the number of rows of the first matrix (S) and the number of columns of the second matrix (T), which is 2x3.
2) To find the pair of matrices that are inverses, we need to check if the product of two matrices results in the identity matrix. Let's check the product of each pair of matrices:
- PR = [3 1] [0 -1/4] = [1 0]
[-4 0] [1 3/4] [-1 1]
The product of matrices PR resulted in the identity matrix, so P and R are inverse matrices.
- QS = [4 4] [7/12 -1/9] = [1 0]
[12 21] [-1/3 1/9] [0 1]
The product of matrices QS resulted in the identity matrix, so Q and S are inverse matrices.
- RT = [0 -1/4] [4 -5 2] = [1 0 0]
[1 3/4] [8 -1 3] [0 1 0]
The product of matrices RT resulted in the identity matrix, so R and T are inverse matrices.
Based on these calculations, the pair of matrices that are inverses are S and R.
3) To find the value of the matrix [15 4] [-6 -2], we simply sum the elements in the matrix. In this case, the value is -6.
4) To evaluate the expression [1 4 -1] [0 3 5], we need to perform matrix multiplication. To do this, we multiply corresponding elements in each row of the first matrix with the corresponding elements in each column of the second matrix, and then sum the products.
The first row of the first matrix [1 4 -1] is multiplied by the first column of the second matrix [0, 0], resulting in (1 * 0) + (4 * 0) + (-1 * 0) = 0.
The second row of the first matrix [1 4 -1] is multiplied by the second column of the second matrix [3, 3], resulting in (1 * 3) + (4 * 3) + (-1 * 3) = 10.
So, the value of [1 4 -1] [0 3 5] is 10.