Exercises

1. Suppose X is a matrix in R. Which of the following is not equivalent to X?
   • A) t( t(X) )
   • B) X %*% matrix(1,ncol(X) )
   • C) X*1
   • D) X%*%diag(ncol(X))

2. Solve the following system of equations using R:

   $$\begin{align*} 3a + 4b - 5c + d &= 10\\ 2a + 2b + 2c - d &= 5\\ a -b + 5c - 5d &= 7\\ 5a + d &= 4 \end{align*}$$

   What is the solution for $$c$?

3. Load the following two matrices into R:

    a <- matrix(1:12, nrow=4)
    b <- matrix(1:15, nrow=3)
   

   Note the dimension of a and the dimension of b.

   In the question below, we will use the matrix multiplication operator in R, %*%, to multiply these two matrices.

   What is the value in the 3rd row and the 2nd column of the matrix product of a and b?

4. Multiply the 3rd row of a with the 2nd column of b, using the element-wise vector multiplication with *.

   What is the sum of the elements in the resulting vector?