Consider the 4\times 3 matrix \boldsymbol{A} defined by A=[100 220 -11a 01b] where a and b are real numbers. What is the smallest value of the rank of \boldsymbol{A} that can be achieved by choosing a and b?
Question
Consider the 4\times 3 matrix \boldsymbol{A} defined by A=[100 220 -11a 01b] where a and b are real numbers.
What is the smallest value of the rank of \boldsymbol{A} that can be achieved by choosing a and b?
Solution
The rank of a matrix is the maximum number of linearly independent rows or columns. In this case, the matrix A is a 4x3 matrix. The maximum rank of a 4x3 matrix is 3, which is the minimum between the number of rows and columns.
However, we can reduce the rank of the matrix by choosing appropriate values for a and b.
If we choose a = 2 and b = 0, the third row becomes a multiple of the second row, and the fourth row becomes a multiple of the first row. This means that the rows are not linearly independent, and the rank of the matrix is reduced.
Therefore, the smallest value of the rank of A that can be achieved by choosing a and b is 2.
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