Question

Simplify: \[\cos \phi \left[ {\begin{array}{*{20}{c}}
  {\cos \phi }&{\sin \phi } \\
  { - \sin \phi }&{\cos \phi }
\end{array}} \right] + \sin \phi \left[ {\begin{array}{*{20}{c}}
  {\sin \phi }&{ - \cos \phi } \\
  {\cos \phi }&{\sin \phi }
\end{array}} \right]\]

Answer 1

Hint: Using the property matrix, that is the scalar multiplication which is a scalar will be multiplied with all the elements inside the matrix. Also remember the basic trigonometric formula as \[{\sin ^2}\phi + {\cos ^2}\phi = 1\]. And also the property of addition of matrix as that each term of both the matrix is added respectively.

Complete step by step answer:

As the given matrix is \[\cos \phi \left[ {\begin{array}{*{20}{c}}
  {\cos \phi }&{\sin \phi } \\
  { - \sin \phi }&{\cos \phi }
\end{array}} \right] + \sin \phi \left[ {\begin{array}{*{20}{c}}
  {\sin \phi }&{ - \cos \phi } \\
  {\cos \phi }&{\sin \phi }
\end{array}} \right]\]
Now, multiplying the terms outside the matrix with the terms inside the matrix as,
\[ = \left[ {\begin{array}{*{20}{c}}
  {{{\cos }^2}\phi }&{\cos \phi \sin \phi } \\
  { - \cos \phi \sin \phi }&{{{\cos }^2}\phi }
\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
  {{{\sin }^2}\phi }&{ - \sin \phi \cos \phi } \\
  {\sin \phi \cos \phi }&{{{\sin }^2}\phi }
\end{array}} \right]\]
And now simply doing the addition of matrix , we get,
\[\left[ {\begin{array}{*{20}{c}}
  {{{\sin }^2}\phi + {{\cos }^2}\phi }&{\cos \phi \sin \phi - \sin \phi \cos \phi } \\
  {\cos \phi \sin \phi - \sin \phi \cos \phi }&{{{\sin }^2}\phi + {{\cos }^2}\phi }
\end{array}} \right]\]
Now, on simplifying the elements by using trigonometric formula as \[{\sin ^2}\phi + {\cos ^2}\phi = 1\], we get
\[ = \left[ {\begin{array}{*{20}{c}}
  1&0 \\
  0&1
\end{array}} \right]\]
Hence, above matrix is the identity matrix of second order.

Note: In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there are other operations that could also be considered in addition to matrices, such as the direct sum and the Kronecker sum.
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix.