How would I answer the following question about the rank of a matrix?

[sktsasus]

"Let A = \begin{pmatrix}1&-8&a\\ \:\:\:4&8a&-16\\ \:\:\:a&32&16\end{pmatrix}


Then the rank of A is ? for a = ?, ? for a = ?, and ? for all other values of a."


I have managed to solve the determinant of the matrix, which came out to be -8(a-8)(a+4)^2 but I did not know how to move forward after this.

Any help on how to proceed will be highly appreciated!

 

[stapel]

\(\displaystyle \mbox{Let }\, A\, =\, \begin{pmatrix}1&-8&a\\ \:\:\:4&8a&-16\\ \:\:\:a&32&16\end{pmatrix}\)

Then the rank of A is ? for a = ?, ? for a = ?, and ? for all other values of a.

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I have managed to solve the determinant of the matrix, which came out to be -8(a-8)(a+4)^2 but I did not know how to move forward after this.

What method did your book teach you for this, that led you to find the determinant of the matrix? "How to move forward after this" point is probably to use that method.

Alternatively, you could use the reduced row echelon form of the actual matrix, and look at the number of non-zero rows. (In this case, you'd been looking for values of "a" that led to non-zero rows, or otherwise.)