8. If A is an nxn square matrix, and I denotes the nxn identity matrix, which of the following statements is not necessarily true? (a) If the columns of A form an orthogonal set, then A is an orthogonal matrix. (b) If AA" = 1, then A is an orthogonal matrix. (c) If A is an orthogonal matrix, its determinant must be +1. (d) If A is an orthogonal matrix, the linear transformation x- Ax preserves lengths an orthogonality.
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
![(b) W2 = W,"
(c) There are two vectors zjin W1 and zzin W 2such that y = z1+ z
(d) y is orthogonal to W, and W2.
5. Let W = Span {u}, where u u =
Let y =
The distance from y to W is
(а) 3
(b)
V3
(c) 9
(d) 1
6.
Which of the following statements is always true?
(а) An n x
(b) An orthogonal matrix is orthogonally diagonalizable.
(c) If P is an n x n matrix with orthogonal columns, then PT = p-1.
(d) Every symmetric matrix is orthogonally diagonalizable.
symmetric matrix has n distinct real eigenvalues.
2
7. Find the orthogonal complement of column space of A.
(a) null space of A
(b) null space of A"
(b) row space of A
(d) row space of A'
8. If A is an nxn square matrix, and I denotes the nxn identity matrix, which of the
following statements is not necessarily true?
(a) If the columns of A form an orthogonal set, then A is an orthogonal matrix.
(b) If AA" = 1, then A is an orthogonal matrix.
(c) If A is an orthogonal matrix, its determinant must be ±1.
(d) If A is an orthogonal matrix, the linear transformation x→ Ax preserves lengths and
orthogonality.
9. A is a Hermitian matrix. Then the possible eigen values of A are
(a) 5, -5
(b)
5, -5i
(c)
5i, -5i
(d)
-5, 5i
10. Let A be a Hermitian matrix. Then, which of the following statements is false?
(a)The diagonal entries of A are all real.
(b) There exists a unitary U such that U'AU is a diagonal matrix.
(c) Eigen values are either real or complex.
(b) If A? = I, then A = I.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98c91c8b-4f64-4d8f-84a9-2f3b63c4d6c4%2F206d37b9-f5d8-415b-a8cd-d4cf304da68d%2Flin0uyk_processed.jpeg&w=3840&q=75)
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