# a. Define the matrices

A = matrix(c(5, -1, 5, -1, 10, 1, 5, 1, -3), nrow = 3, byrow = TRUE)
B = matrix(c(5, 3, -2, 1, 1, 0), nrow = 2, byrow = TRUE)
C = matrix(c(2, -3, 1, 4), nrow = 2, byrow = TRUE)
x = matrix(c(5, 1, 4), nrow = 3, byrow = FALSE)

# b.Compute BA

BA = B %*% A
BA

##      [,1] [,2] [,3]
## [1,]   12   23   34
## [2,]    4    9    6

# c.Matrix multiplication that doesn't work

tryCatch({
  AB = A %*% B  # This multiplication will fail
  print(AB)      # If no error occurs (unexpected), print the result
}, error = function(e) {
  print("Matrix multiplication AB is not possible due to incompatible dimensions.")
  print(e)  # Optional: Display the error message for debugging
})

## [1] "Matrix multiplication AB is not possible due to incompatible dimensions."
## <simpleError in A %*% B: non-conformable arguments>

# d.Compute inner product

x_T_x = t(x) %*% x
x_T_x

##      [,1]
## [1,]   42

# e.Solve A.y = x

y = solve(A, x)
y

##           [,1]
## [1,] 0.8665049
## [2,] 0.1699029
## [3,] 0.1674757

# f.Compute L2 norm of A

L2_norm_A = max(svd(A)$d)
L2_norm_A

## [1] 10.19274

# g.Compute the 2-condition number of A

condition_number = max(svd(A)$d) / min(svd(A)$d)
condition_number

## [1] 1.847177