Multiply the Matrices

Problem

[[cos(b),i/g*sin(b)],[i*g*sin(b),cos(b)]]*[[cos(a),i/d*sin(a)],[i*d(sin(a)),cos(a)]]

Solution

1. Identify the matrix multiplication rule for 2×2 matrices, where the element in row i and column j is the dot product of row i of the first matrix and column j of the second matrix.

2. Calculate the top-left element (c_11) by multiplying the first row of the first matrix by the first column of the second matrix.

(c_11)=cos(b)*cos(a)+(i/g*sin(b))*(i*d(sin(a)))

(c_11)=cos(b)*cos(a)−d/g*sin(b)*sin(a)

3. Calculate the top-right element (c_12) by multiplying the first row of the first matrix by the second column of the second matrix.

(c_12)=cos(b)*(i/d*sin(a))+(i/g*sin(b))*cos(a)

(c_12)=i*(1/d*cos(b)*sin(a)+1/g*sin(b)*cos(a))

4. Calculate the bottom-left element (c_21) by multiplying the second row of the first matrix by the first column of the second matrix.

(c_21)=(i*g*sin(b))*cos(a)+cos(b)*(i*d(sin(a)))

(c_21)=i*(g*sin(b)*cos(a)+d(cos(b))*sin(a))

5. Calculate the bottom-right element (c_22) by multiplying the second row of the first matrix by the second column of the second matrix.

(c_22)=(i*g*sin(b))*(i/d*sin(a))+cos(b)*cos(a)

(c_22)=−g/d*sin(b)*sin(a)+cos(b)*cos(a)

Final Answer

[[cos(b),i/g*sin(b)],[i*g*sin(b),cos(b)]]*[[cos(a),i/d*sin(a)],[i*d(sin(a)),cos(a)]]=[[cos(b)*cos(a)−d/g*sin(b)*sin(a),i*(1/d*cos(b)*sin(a)+1/g*sin(b)*cos(a))],[i*(g*sin(b)*cos(a)+d(cos(b))*sin(a)),cos(b)*cos(a)−g/d*sin(b)*sin(a)]]

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