What value of p for which A = [p 3 3; 3 p 3; 3 3 p] is of rank 1? If A is rank 1 matrix, then its determinant will also be 0.
Understand the Problem
The question is asking for the value of 'p' that makes the matrix A have a rank of 1. It also notes that if the rank is 1, then the determinant of matrix A will be 0. To solve this, we would typically set the determinant of the matrix equal to 0 and solve for 'p'.
Answer
The value of \( p \) is \( 3 \).
Answer for screen readers
The value of ( p ) that makes the matrix ( A ) have a rank of 1 is ( p = 3 ).
Steps to Solve
- Write down the determinant formula for matrix A
The determinant of matrix $A$ can be calculated using the formula for a $3 \times 3$ matrix: $$ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) $$
For our matrix: $$ A = \begin{pmatrix} p & 3 & 3 \ 3 & p & 3 \ 3 & 3 & p \end{pmatrix} $$
- Calculate the determinant of matrix A
Applying the determinant formula:
- For $a = p$, $b = 3$, $c = 3$, $d = 3$, $e = p$, $f = 3$, $g = 3$, $h = 3$, and $i = p$, we get: $$ |A| = p(p^2 - 9) - 3(3p - 9) + 3(9 - 3p) $$
- Simplify the determinant expression
Simplifying the expression we derived: [ |A| = p^3 - 9p - 9p + 27 + 27 - 9p ] Combine like terms: $$ |A| = p^3 - 27p + 54 $$
- Set the determinant to zero
To find the value of $p$ that makes the rank of $A$ equal to 1, we set the determinant to 0: $$ p^3 - 27p + 54 = 0 $$
- Solve the polynomial equation for p
We can try factoring or using the rational root theorem or numerical methods to find the roots.
By testing simple values, we can find that $p = 3$ satisfies the equation.
The value of ( p ) that makes the matrix ( A ) have a rank of 1 is ( p = 3 ).
More Information
A matrix of rank 1 means that all rows (or columns) are linearly dependent, and thus the determinant must equal zero. The roots of the resultant cubic equation give the possible values for ( p ).
Tips
- Miscalculating the determinant of a matrix: It's vital to apply the correct formula systematically.
- Forgetting to set the determinant to zero: This is essential for finding the required value of ( p ).
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