Evaluate each determinant.

Question

∣ \begin{matrix} 1 & - 2 & 3 \\ 0 & 0 & 0 \\ 1 & 10 & - 12 \end{matrix} ∣

Accepted Answer

Hello, everyone. We are going to find the value of the determinant given below. We are given a three by three matrix with the first row reading four negative 79. The second row reading 000 in the third row of reading 3 20 negative 28. Our answer choices are a zero B negative 30 C negative D not defined first, I'm going to make a three by three matrix with generic letters. So my first row will be ABC second row D E F third row G H I, this way I can find the placement for my determinant. So to find the determinant of this matrix, I would do the value of a multiplied by the determinant of the minor matrix which is a two by two created by the rows and columns that do not include A. So this would be E for the first row E F second row H I then minus the value of B multiplied by the determinant of the two by two matrix created by the rows and columns, not including B so D F for the first row and G I for the second row plus the value of C multiplied by the determinant of the two by two matrix D E for the first row G H for the second row. So now that we have the format, let's try it with the numbers, we're given a, here is four and that's gonna be multiplied by the determinant. And I'm gonna actually make this into a matrix, sorry, the determinant of E F so 00 and the second row would be H I which would be 20 and negative minus our value of B which is negative seven multiplied by the determinant of our two by two matrix. That would be made up of the values that do not have negative seven. So 00 is our first row. Three, negative 28 is our second row plus the value of C which is nine multiplied by the determinant of the two by two matrix. That would be 00 for the first row 3 20. For the second row recall that define the determinant of a two by two matrix. We multiply diagonally starting at the top left and then subtract the products. So we would have four multiplied by and it would be zero multiplied by negative 28 which is zero minus zero, multiplied by 20 which is zero, close parentheses. Um double negative is like adding. So I'm gonna write this as plus seven multiplied by the determinant. So it would be zero multiplied by negative 28 which is zero minus three, multiplied by zero, which is zero plus nine, multiplied by the determinant of that two by two matrix. So zero times 20 is zero minus three, multiplied by zero is zero closed parentheses. And it looks like all of our parentheses come out to zero. So we would have four multiplied by zero plus seven, multiplied by zero plus nine, multiplied by zero. And any value multiplied by zero is zero. So we have zero plus zero plus zero and this will equal zero. So the determinant of this three by three matrix is zero, which is answer choice. A have a nice day.

Evaluate each determinant.
$∣ \begin{matrix} 1 & - 2 & 3 \\ 0 & 0 & 0 \\ 1 & 10 & - 12 \end{matrix} ∣$

Key Concepts

Determinant of a Matrix

Methods for Calculating Determinants

Properties of Determinants

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