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Matrix Algebra Worksheet 1 Solutions
- (a) Write a 2 × 4 matrix.
Solution: Here’s one possible answer: [ 1 2 3 4 1 0 0 2
]
(b) Write a 4 × 2 matrix.
Solution: Here’s one possible answer:
(c) Multiply the two matrices that you just wrote.
Solution: The problem does not state which order to multiply the matrices in (and either order is valid here) so let’s just pick an order.
[
]
[
]
[
]
If we had multiplied the matrices in the other order, we would have instead ended up with a 4 × 4 matrix.
(d) Write two matrices that you can’t multiply.
Solution: There are many possible answers here. We just need two matrices so that the number of columns of the first matrix is not equal to the number of rows in the second. One example is
and
(e) Write two matrices that you can multiply in one order but not in the other order.
Solution: We need two matrices so that the number of columns of the first one is not equal to the number of rows of the second, but the number of columns of the second is equal to the number of rows of the first. One example is
(^) and
(f) Write a matrix that is equal to its transpose.
Solution: First observe that if a matrix has dimensions n × m then its transpose has dimensions m×n. So if the matrix is equal to its transpose, we must have n = m—i.e. the matrix must be square. Furthermore, since the rows of the transpose are just the columns of the original matrix, we must find a matrix whose first column is the same as its first row, second column is the same as its second row, and so on. So one way to find such a matrix is as follows: First fill in the first row and then fill in the first column to match it. Then fill in the remainder of the second row and fill in the second column to match it, and so on. One example of such a matrix is
(g) Write a square matrix that is not invertible.
Solution: We need a matrix whose determinant is zero. One such matrix is [ 0 0 0 0
]
Another example is (^) [ 4 2 6 3
]
(h) Write a square matrix that is invertible.
Solution: We need a matrix whose determinant is not zero. One example is [ 1 0 0 1
]
Another one is (^) [ 1 2 3 4
]
triangle is zero. A few possible answers:
[ 1 3 0 1
] [
] [
]
(m) Write a system of linear equations with infinitely many solutions.
Solution:
x + y = 1 2 x + 2y = 2
(n) Write a system of linear equations with exactly one solution.
Solution:
x = 1 y = 2
(o) Write a system of linear equations with no solution.
Solution:
x + y = 1 2 x + 2y = 3
- True or false:
(a) The following matrix is diagonal (^)
Solution: True. All entries not on the diagonal are zero.
(b) The following matrix is upper triangular
Solution: True. All entries not on the upper triangle are zero.
(c) If the determinant of a square matrix is not zero, then the matrix is invertible.
Solution: True. This was mentioned in lecture.
(d) If the coefficient matrix is not invertible then a system of linear equations cannot have a solution.
Solution: False. For instance, consider the following system of linear equations
x + y = 1 2 x + 2y = 2
There is clearly a solution (in fact, there are infinitely many solutions) but the coef- ficient matrix is (^) [ 1 1 2 2
]
which is not invertible.
- Find all solutions of the following system of linear equations.
4 x 2 + 8 x 3 = 12 x 1 − x 2 + 3 x 3 = − 1 3 x 1 − 2 x 2 + 5 x 3 = 6
Solution: First, let’s write the corresponding augmented matrix.
Now we can use Guassian elimination
(^) −Switch−−−−^ −R−^1 −and−−^ R−→^2
−R−^3 −−−^3 −R−^1 →−−R→^3
(^14) R 2 →R 2 −−−−−−→
−R−^3 −−−R−^2 →−−R→^3
− 16 R 3 →R 3 −−−−−−−→
