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Given π΄ is equal to four, negative seven, negative three, negative four, find its multiplicative inverse if possible.
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Remember, for a two-by-two matrix π΄ which is equal to π, π, π, π, its inverse is given by the formula one over the determinant of π΄ multiplied by π, negative π, negative π, π, where the determinant of π΄ is found by subtracting the product of the top right and the bottom left element from the product of the top left and the bottom right.
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Thatβs π multiplied by π minus π multiplied by π.
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Notice this means if the determinant of the matrix π΄ is zero, the inverse cannot exist since one divided by the determinant of π΄ becomes one divided by zero, which is undefined.
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Letβs substitute the values for our matrix into the formula for the determinant of π΄: π is four, π is negative seven, π is negative three, and π is negative four.
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So the determinant of π΄ is π multiplied by π, thatβs four multiplied by negative four, minus π multiplied by π, thatβs negative seven multiplied by negative three.
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Four multiplied by negative four is negative 16, and negative seven multiplied by negative three is just 21.
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So this becomes negative 16 minus 21, which is negative 37.
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Since the determinant of π΄ is not zero, the inverse of π΄ does exist.
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And now that we have that determinant, we can substitute everything we know into the formula for the inverse of π΄.
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One over the determinant of π΄ is one over negative 37 or negative one thirty-seventh.
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Then we switch the value for π and π.
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So we get negative four in the top left corner and four in the bottom right.
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We multiply the elements that we named π and π by negative one.
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And that gives us seven in the top right corner and three in the bottom left.
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The inverse of π΄ is therefore negative one thirty-seventh multiplied by negative four, seven, three, four.