MTH501 Final Term Solved MCQs Set 3

from book

A square matrix A is said to be diagonalizable if A is similar to a diagonal matrix. i.e. if A = -1 PDP for some invertible matrix P and some diagonal matrix D.

Q8. The inverse of an invertible lower triangular matrix is

1. lower triangular matrix
2. upper triangular matrix
3. diagonal matrix

The standard operations on triangular matrices conveniently preserve the triangular form: the sum and product of two upper triangular matrices is again upper triangular. The inverse of an upper triangular matrix is also upper triangular, and of course we can multiply an upper triangular matrix by a constant and it will still be upper triangular. This means that the upper triangular matrices form a subalgebra of the ring of square matrices for any given size. The analogous result holds for lower triangular matrices. Note, however, that the product of a lower triangular with an upper triangular matrix does not preserve triangularity.

Q9. If P is a parallelepiped in R3, then

{volume of T (P)} = |detA|. {volume of P}

1. Where T is determined by a 2 2

matrix A

1. Where T is determined by a 2 3matrix A
2. Where T is determined by a 3 3 matrix A

Correct answer: Where T is determined by a 3 3 matrix A

not conf.

1. Where T is determined by a 3 2 matrix A

Q10. Let A be a n m matrix of rank r

then row space of A has dimension

1. m  r
2. n  r
3. r
4. nm

Main article: Rank (linear algebra) The dimension of the row space is called the rank of the matrix. This is the same as the maximum number of linearly independent rows that can be chosen from the matrix. For example, the 3 × 3 matrix in the example above has rank two. The rank of a matrix is also equal to the dimension of the column space. The dimension of the null space is called the nullity of the matrix, and is related to the rank by the following equation:

where n is the number of columns of the matrix A. The equation above is known as the rank-nullity theorem.

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Q11. P4

The dimension of the vector space is

1. 4
2. 3
3. 5
4. 1