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🔢Matrices

4 min read•december 13, 2021

Jenni MacLean

Amrita Arora

3x - y = 7 and 2x + y = 8 become

You'll notice throughout this article matrices with different types of brackets: some enclosed in parentheses and others in square brackets. Functionally, it makes no difference which one you use. They're just different notations that certain professions are partial towards.

Finding the inverse of a matrix can be helpful when you want to get from any matrix to the Identity Matrix, which is the matrix representation of a whole (1):

**Inverses of Matrices **

The **inverse of a matrix**, usually represented by **A⁻¹, **is equal to the **adjoint **of a matrix divided by the **determinant**. We can find the **adjoint **of a matrix by switching *a* and *d* and making *b* and *c* negative in a standard 2x2 matrix:

The inverse of a matrix, usually represented by A⁻¹, is equal to the adjoint of a matrix divided by the determinant. We can find the adjoint of a matrix by switching a and d and making b and c negative in a standard 2x2 matrix:

For the inverse of a matrix to exist, it must be a **square matrix 🟦**. In a square matrix, the number of terms counted vertically must equal the number of terms counted horizontally; in other words, it must be 2x2, 3x3, etc., not 3x2 or 3x4.

When writing down the dimensions of a matrix, the first number represents the number of rows, and the second number represents the number of columns. For example, in a 2x3 matrix, the matrix has 2️⃣rows and 3️⃣columns. In addition, the determinant of the matrix must be greater than 0. So, if you calculate the determinant of a matrix and it equals 0, the inverse of that matrix does not exist **❌**

**The Determinant of a 2x2 Matrix**

det(A) = |A| = (*ad*-*bc*)

The determinant can be represented by “det(A)” or more typically, |A|, with absolute value symbols.

Let’s take a look at the 2x2 matrix in the problem below 🔎

To find the determinant, we must calculate the product of *ad* and *bc. *Using the previous *abcd* matrix, we can say a = 3, b = 4, c = 2, and d = -5. Thus, *ad* = -15 and *bc *= 8. If we substitute -15 and 8 for *ad* and *bc* in the determinant formula *ad-bc*, we get (-15 - 8) = -23. Thus, the determinant of this matrix is -23.

**The Determinant of a 3x3 Matrix**

|A| =* a(ei − fh) − b(di − fg) + c(dh − eg) *where

Let’s take a look at the 3x3 matrix supplied in the problem below 🔎

We’ll consider the bottom 6 terms of the **matrix **as 3 different subcolumns. The formula requires that you multiply *a* by the **determinant **of the two rightmost *subcolumns *(efhi), *b* by the determinant of the outer two *subcolumns *(dgfi), and *c* by the **determinant **of the two rightmost *subcolumns *(dgeh).

If we substitute the given matrix’s values into this formula, we have the following 🔢:

|A| = 2(-2-6) - 3(1-9) + 1(-2-6)

|A| = 2(-8) - 3(-8) + 1(-8)

|A| = -16 + 24 -8

|A| = 0

The determinant of Matrix A is 0. This means that its inverse does not exist ❌.

**Finding the Determinant on the TI-84 🖩**

- Power on your calculator using the button on the bottom right.
- Press the
*2nd*button, then the*X-1*button to get to the matrix menu. - Use the right arrow key to move from NAMES to EDIT.
- Use the arrow keys to navigate to the letter you want your matrix to represent and press
*enter*(bottom right of your calculator). - Set the dimensions of your matrix and press the down key after the second dimension to enter the matrix.
- Enter your matrix values. Go back to the
*matrix*menu as in Step 2. - Use the right arrow key to navigate from NAMES to MATH.
- Choose option 1: "det(" and press enter.
- Return to the
*matrix*menu, select the letter of the matrix you chose earlier and press*enter*. - Close the parentheses and press
*enter*again to get the determinant.

On the ACT, you may need to calculate the product or determinant of a matrix. Check out Fiveable's __ACT Math Section Review__ under our ACT Crams section for more on that!

On the SAT, you may need to add, subtract, multiply, or divide matrices in addition to finding their determinants. Luckily, Fiveable also has resources for SAT Math - see our __SAT Crams__ for more. Overall, general matrix operations are fair game in these standardized tests.

Generally, you can use a test-approved graphing calculator to solve these questions like the TI-84. Matrices may also appear on the __Calculus AB__ test.

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