6 points] a) Suppose that A is a 6 x 4 matrix. What is the largest possible rank
ID: 3114857 • Letter: 6
Question
6 points] a) Suppose that A is a 6 x 4 matrix. What is the largest possible rank of A? What is the smallest possible dimension of ker(A)? Give an example of an A that achieves the bounds you find (if it isn't "obvious" that the A you give works, then you need to show it does). b) Suppose A is a 4 x 6 matrix. What is the largest possible rank of A? What is the smallest possible dimension of ker(A)? Give an example of an A that achieves the bounds you find. (if it isn't "obvious" that the A you give works, then you need to show it does). c) Suppose that A is an rn x n matrix with rank 5 and dim ker(A)-4. What are the possible sizes of A? (i.e. the possible values of m and n d) Show that any 2 x 2 singular, nonzero matrix must have rank 1Explanation / Answer
(a) If the matrix A has 4 columns and 6 rows, the maximum number of pivots is 4. Thus, the largest possible rank of A is 4. Now, as per the Rank-Nullity theorem, the smallest possible dimension of Ker(A) is no. of columns of A-maximum possible rank of A = 4-4 = 0. Let A =
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Then Rank(A) is 4 and dim(Ker(A)) = 0 as the equation AX = 0 has only the trivial solution.
(b) If the matrix A has 6 columns and 4 rows, the maximum number of non-zero rows in its REEF is 4. Thus, the largest possible rank of A is 4. Now, as per the Rank-Nullity theorem, the smallest possible dimension of Ker(A) is no. of columns of A-maximum possible rank of A = 6-4 = 20. Let A =
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If X = (x1,x2,x3,x4,x5,x6)T, then the equation AX = 0 is equivalent to x1 +x6 = 0, x2 +x5 =0
, x3 +x6 = 0 and x4 +x5 = 0. Now, let x5 = r and x6 = t. Then X = (-t,-r,-t,-r,r,t)T=r(0,-1,0,-1,1,0)T+ t(-1,0,-1,0,0,1)T so that Ker(A) = span{(0,-1,0,-1,1,0)T,(-1,0,-1,0,0,1)T } and dim(Ker(A)) =
(c ) If a mxn matrix has rank 5 and dim(Ker(A) ) = 4, then as per the Rank-Nullity theorem, the number of columns has to be 5+4 = 9.Also, since the rank is 4, the number of non-zero rows in the RREF of A is 5 so that m 5. Thus, the size of the matrix is mx9, where m 5.
(d) If A is a singular non-zero matrix, then A is of the form (as we must have det A=0)
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where a and b are arbitrary real numbers. The RREF of A will be
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Hence Rank (A) = 1.
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