Determinants - GRE Subject Test: Math

Card 0 of 5

Question

Given the following matrix, find the determinant, if possible.

$A=\begin{bmatrix} -4&4 \ 4&-4 \end{bmatrix}$

Answer

Write the formula to find the determinant given a 2 by 2 matrix.

$\begin{bmatrix} a&b \ c&d \end{bmatrix}=ad-bc$

Substituting in the given matrix we are able to find the determinant.

$\begin{bmatrix} -4&4 \ 4&-4 \end{bmatrix}=(-4)(-4)-(4)(4)=16-16=0$

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Question

Evaluate the determinant of the following matrix.

$\begin{bmatrix} 5& -3\ 7& 8 \end{bmatrix}$

Answer

Remember, to evaluate the determinant of a matrix use the following:

$\begin{vmatrix} a&b \ c&d \end{vmatrix}=ad-bc$

The first step would be to write the determinant of the matrix:

$\begin{vmatrix} 5& -3\ 7&8 \end{vmatrix}$

Now we can evaluate:

$\begin{vmatrix} 5& -3\ 7&8 \end{vmatrix}=(5)(8)- (-3)(7)=40+21=61$

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Question

Evaluate the determinant of the following matrix.

$\begin{bmatrix} 2& 4& 3\ 3& 0& 1\ 1& 2& 5 \end{bmatrix}$

Answer

To find the determinant of a 3 x 3 matrix, we must use the following:

$\begin{vmatrix} a& b& c\ d& e& f\ g& h&k \end{vmatrix}=a\begin{vmatrix} e& f\ h& k \end{vmatrix} -b\begin{vmatrix} d& f\ g& k \end{vmatrix} + c\begin{vmatrix} d& e\ g& h \end{vmatrix}$

The first thing we must do is write the determinant:

$\begin{vmatrix} 2& 4& 3\ 3& 0& 1\ 1& 2& 5 \end{vmatrix}$

Now we can proceed to evaluate the determinant

$\begin{vmatrix} 2& 4& 3\ 3& 0& 1\ 1& 2& 5 \end{vmatrix}=2\begin{vmatrix} 0& 1\ 2& 5 \end{vmatrix}-4\begin{vmatrix} 3&1 \ 1& 5\end{vmatrix}+3\begin{vmatrix} 3& 0\ 1&2 \end{vmatrix}$

Notice that the numbers 2, 4, and 3 are being multiplied by the determinants of the 2x2 matrices so we have:

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Question

Find the determinant of the matrix: $\begin{vmatrix} -4 & -7 \ 1 & 2 \end{vmatrix}$

Answer

Step 1: We need to recall how to find the determinant of a matrix. To find the determinant of a $2\ast2$ matrix, we need to use the equation , where =Determinant, and are from the matrix.

Step 2: Identify a,b,c, and d in the original matrix.
a=first number on top row, b=second number on top row (next to a), c=first number on the bottom row, and d is the second number on the bottom row (next to c).

In this matrix, a=, b=, c=, and d=.

Step 3: Substitute the values of a,b,c, and d into the equation to find the determinant of the matrix.

$ad-bc=(-4\cdot 2)-(-7\cdot 1)$ We will simplify the right side.

. We see that there are two negative signs in the middle, which will become a plus sign.

. Simplify the right side.

. is the determinant.

The determinant of the matrix is .

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Question

Find the determinate of Matrix A.

$A=\begin{bmatrix} 8&2 \ 5& 3 \end{bmatrix}$

Answer

Matrix A is given below.

$A=\begin{bmatrix} a&b \ c& d \end{bmatrix}$

The formula for the determinate of a 2x2 matrix is:

Plugging in the values gives us:

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