A sample matrix inversion in PHP using Gauss-Jordan elimination. This is implementation is meant for clarity, not for performance, memory usage, or numerical stability. You can optionally turn on the debug flag to output matrix state after every iteration. See https://en.wikipedia.org/wiki/Gauss-Jordan_elimination for more info. Enjoy!

test-matrix-invert.php

<?php

/**
 * Inverts a given matrix
 *
 * @param array $A matrix to invert
 * @param boolean $debug whether to print out debug info
 *
 * @return array inverted matrix
 */
function invert($A, $debug = FALSE)
{
	/// @todo check rows = columns

	$n = count($A);

	// get and append identity matrix
	$I = identity_matrix($n);
	for ($i = 0; $i < $n; ++ $i) {
		$A[$i] = array_merge($A[$i], $I[$i]);
	}

	if ($debug) {
		echo "\nStarting matrix: ";
		print_matrix($A);
	}

	// forward run
	for ($j = 0; $j < $n-1; ++ $j) {
		// for all remaining rows (diagonally)
		for ($i = $j+1; $i < $n; ++ $i) {
			// if the value is not already 0
			if ($A[$i][$j] !== 0) {
				// adjust scale to pivot row
				// subtract pivot row from current
				$scalar = $A[$j][$j] / $A[$i][$j];
				for ($jj = $j; $jj < $n*2; ++ $jj) {
					$A[$i][$jj] *= $scalar;
					$A[$i][$jj] -= $A[$j][$jj];
				}
			}
		}
		if ($debug) {
			echo "\nForward iteration $j: ";
			print_matrix($A);
		}
	}

	// reverse run
	for ($j = $n-1; $j > 0; -- $j) {
		for ($i = $j-1; $i >= 0; -- $i) {
			if ($A[$i][$j] !== 0) {
				$scalar = $A[$j][$j] / $A[$i][$j];
				for ($jj = $i; $jj < $n*2; ++ $jj) {
					$A[$i][$jj] *= $scalar;
					$A[$i][$jj] -= $A[$j][$jj];
				}
			}
		}
		if ($debug) {
			echo "\nReverse iteration $j: ";
			print_matrix($A);
		}
	}

	// last run to make all diagonal 1s
	/// @note this can be done in last iteration (i.e. reverse run) too!
	for ($j = 0; $j < $n; ++ $j) {
		if ($A[$j][$j] !== 1) {
			$scalar = 1 / $A[$j][$j];
			for ($jj = $j; $jj < $n*2; ++ $jj) {
				$A[$j][$jj] *= $scalar;
			}
		}
		if ($debug) {
			echo "\n1-out iteration $j: ";
			print_matrix($A);
		}
	}

	// take out the matrix inverse to return
	$Inv = array();
	for ($i = 0; $i < $n; ++ $i) {
		$Inv[$i] = array_slice($A[$i], $n);
	}

	return $Inv;
}

/**
 * Prints matrix
 *
 * @param array $A matrix
 * @param integer $decimals number of decimals
 */
function print_matrix($A, $decimals = 6)
{
	foreach ($A as $row) {
		echo "\n\t[";
		foreach ($row as $i) {
			echo "\t" . sprintf("%01.{$decimals}f", round($i, $decimals));
		}
		echo "\t]";
	}
}

/**
 * Produces an identity matrix of given size
 *
 * @param integer $n size of identity matrix
 *
 * @return array identity matrix
 */
function identity_matrix($n)
{
	$I = array();
	for ($i = 0; $i < $n; ++ $i) {
		for ($j = 0; $j < $n; ++ $j) {
			$I[$i][$j] = ($i == $j) ? 1 : 0;
		}
	}
	return $I;
}

$A = array(
	array( 10, -15,  30,   6, -8 ),
	array(  0,  -4,  60,  11, -5 ),
	array(  8,   9,   2,   3,  7 ),
	array( 25,  10,  -9,   9,  0 ),
	array( 13,   3, -12,   5,  2 ),
);

echo "\nMatrix:";
print_matrix($A);
echo "\n";

$B = invert($A);

echo "\nInversion result:";
print_matrix($B);
echo "\n\n";

?>

Crashes on division by zero for:
$A = [
[1, 0, 7],
[0, 2, 0],
[0, 0, 3],
];