Solve each equation.

Question

Matrix equation with variables and constants set equal to an expression, prompting to solve each equation.

Accepted Answer

Hello, everyone. We have been asked to work out the given equation and the equation is a three by three matrix where the top row is 135 X, the middle row is 021. The third row is negative 12 X and it equals two. Our answer choices are a 1/ B four C 7/ D 12 7th. I am going to find the determinant of this three by three matrix and simplify as much as I can before setting it equal to two. I recall that to find the determinant of a three by three matrix, I'm going to multiply each element from the top row by a minor matrix. So in this case, I'm gonna start with one and I'm gonna multiply that by, it's gonna be the determinant of the matrix comprised of the rows and columns, not including the element one. So this minor matrix is gonna be top row 21 2nd row two X and then we subtract the element three. So the second element in the top row, and we're going to multiply that by the determinant of the matrix 01 for the top row negative one X for the second row. And again, that's comprised of the values that do not include the element three. And then plus five X, which is the third element in the top row, which will be multiplied by the determinant. Again, of the matrix created excluding the rows and columns including five X. So 02 is my top row, negative 12 is my second row. And all of this will have to equal two for us to finding the value of X. So next, I'm gonna take the value of one and I'm gonna multiply it in parentheses by the determinant from this first minor matrix. So we're called to find the determinant of a two by two matrix. We first find the product of the diagonal starting at the top left. So in this case, two multiplied by X is two X and then we subtract the product of the other diagonal. So one multiplied by two, which is two close parentheses and then minus three multiplied by the parentheses. Zero, multiplied by X which is zero minus one multiplied by negative one, which is a negative one. So double negative is like adding. So plus one closed parentheses plus five X multiplied by the parentheses, zero, multiplied by two, which is zero minus two, multiplied by negative one, which is negative two. And again, double negatives are like adding. So I'm gonna have zero plus two in my parentheses and this equals two. Now I'm gonna simplify what's in the parentheses and distribute as needed. So for my first set, I have one multiplied by the parentheses. Two X minus two. I can rewrite that as just two X minus two. Next I have negative three multiplied by the parentheses. Zero plus one, zero plus one is one, one multiplied by negative three is negative three. Next five X multiplied by the parentheses. Zero plus two zero plus two is 22, multiplied by five, X is 10 X and this still has to equal two. So we now have the equation two, X minus two minus three plus 10 X equals two. I'm gonna combine my light terms that are on the same side. So I'll have 12 minus five equals two. Solving for X. I'm gonna add five to both sides. I have 12 X equals seven dividing by 12. X has to equal 7/12. That should be our answer. I believe it is. And that matches answer choice. C 7/12 have a nice day.

Solve each equation.

Key Concepts

Understanding Equations

Isolating the Variable

Interpreting and Correcting Equation Format

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