𝑆𝑢𝑝𝑝𝑜𝑠𝑒 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙 𝑜𝑓 𝑠𝑜𝑚𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝐴 𝑖𝑠 𝑓𝑜𝑢𝑛𝑑 𝑡𝑜 𝑏𝑒 𝑝(𝜆) = (𝜆 - 1)𝜆 -32  𝜆 -43  . 𝑊ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑖𝑧𝑒 𝑜𝑓 𝐴?

1. Break Down the Problem

To solve for the size of the matrix $A$ based on its characteristic polynomial $p(\lambda)$, we need to understand the relationship between the degree of the polynomial and the size of the matrix.

2. Relevant Concepts

The characteristic polynomial of a matrix $A$ is given by $p(\lambda) = \det(A - \lambda I)$, where $I$ is the identity matrix of the same size as $A$. The degree of the characteristic polynomial corresponds to the size (order) of the matrix.

3. Analysis and Detail

The provided characteristic polynomial is:

$p(\lambda) = (\lambda - 1)(\lambda - 32)(\lambda - 43)$

To find the degree of this polynomial, we can expand it:

• The factors are three linear terms: $(\lambda - 1)$, $(\lambda - 32)$, and $(\lambda - 43)$.
• The total number of factors (terms) indicates the degree of the polynomial.

4. Verify and Summarize

Since there are three factors, the degree of the polynomial is 3. Thus, the size of the matrix $A$ is $3 \times 3$.

Final Answer

The size of matrix $A$ is $3 \times 3$.