Let a and b be real numbers. Find all possible values of the rank and nullity of
A =
⎛
⎜
⎝
2 3 a b
1 2 3 4
4 3 2 1
⎞
⎟
⎠

Question

Let a and b be real numbers. Find all possible values of the rank and nullity of
A =
⎛
⎜
⎝
2 3 a b
1 2 3 4
4 3 2 1
⎞
⎟
⎠

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the rank of the matrix $A = \begin{pmatrix} 2 & 3 & a & b \ 1 & 2 & 3 & 4 \ 4 & 3 & 2 & 1 \end{pmatrix}$, we need to determine the number of linearly independent rows or columns. ‖ ‖ step 2 ⋮ We perform row reduction on the matrix $A$. The row-reduced form will help us identify the number of non-zero rows, which corresponds to the rank. ‖ ‖ step 3 ⋮ After row reduction, we find that the matrix can be reduced to a form where the number of non-zero rows is 3, indicating that the rank of the matrix is 3. ‖ ‖ step 4 ⋮ The nullity of a matrix is given by the formula $ ext{nullity} = n - ext{rank}$, where $n$ is the number of columns. For matrix $A$, $n = 4$. ‖ ‖ step 5 ⋮ Using the formula, $ ext{nullity} = 4 - 3 = 1$. Therefore, the nullity of the matrix $A$ is 1. ‖ ∻Answer∻ ⚹ The rank of the matrix $A$ is 3, and the nullity is 1. ⚹ ∻Key Concept∻ ⚹Rank and Nullity of a Matrix⚹ ∻Explanation∻ ⚹The rank of a matrix is the number of linearly independent rows or columns, and the nullity is the number of solutions to the homogeneous equation $Ax = 0$. For a matrix with $n$ columns, the nullity is given by $n - ext{rank}$.⚹◊What is the relationship between the rank and nullity of a matrix, and how can it be determined for matrix A⍭ Generate me a similar question◊

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