The largest possible rank of an m by n matrix is
the minimum value of m and n
. The largest possible rank gives the smallest possible nullity. (a) Since A is 4 by 4 the largest possible rank is 4, the smallest possible nullity is 0.
What is the smallest possible rank of a matrix?
When the rank equals the smallest dimension it is called “full rank”, a smaller rank is called “rank deficient”. The rank is at least 1, except for a zero matrix (a matrix made of all zeros) whose rank is
0
.
What is the maximum possible rank of a matrix?
The correct answer is (C). Since the matrix has more than zero elements, its rank must be greater than zero. And since it has fewer rows than columns, its maximum rank is
equal to the maximum number of linearly independent rows
.
What is the smallest possible dimension of Nul A?
Final Answer:
Zero
is the smallest possible dimension of the null space of a 6 × 4 matrix A.
What is the largest possible dimension that the row space could have?
Hence, the largest possible dimension of the row space of A is
4
. Also, the largest possible dimension of the null space is also 4 because Nul A is a subspace of R4.
What is the rank of a 3×3 matrix?
You can see that the determinants of each 3 x 3 sub matrices are equal to zero, which show that the rank of the matrix is not 3. Hence, the rank of the matrix
B = 2
, which is the order of the largest square sub-matrix with a non zero determinant.
Can a matrix have rank 0?
Only a zero matrix has rank zero
. f is injective (or “one-to-one”) if and only if A has rank n (in this case, we say that A has full column rank). … If A is a square matrix (i.e., m = n), then A is invertible if and only if A has rank n (that is, A has full rank).
What is range of matrix?
In linear algebra, the column space (also called the range or image) of a matrix A is
the span (set of all possible linear combinations) of its column vectors
. The column space of a matrix is the image or range of the corresponding matrix transformation.
What is the definition of rank of matrix?
:
the order of the nonzero determinant of highest order that may be formed from the elements of a matrix by selecting arbitrarily an equal number of rows and columns from it
.
What are the smallest possible dimensions?
The smallest possible size for anything in the universe is the Planck Length, which is
1.6 x10
– 35
m across
.
What is the dimension of Nul A?
The dimension of Nul A is
the number of free variables in Ax = 0 (ie the number of columns without pivots in A)
, and the dimension of Col A is the number of pivot columns. Definition: The rank of A is the dimension of the column space of A (ie.
Is W in Nul a calculator?
Yes,
the vector “w” is in Nul A
. A basis or spanning set for Nul A are these two vectors: , .
What is dim Row A?
dim (Col A) =
rankA
.
The dimension of the column space of a matrix equals the rank of the matrix
.
Is Col an AR 4?
No
, Col A is not R4. Since A has four pivot columns, dim Col A is 7.
