Matrix Operations
Adding, subtracting, and scalar multiplication of matrices; reading matrix dimensions — ISEE Upper only
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Imagine you are keeping track of video game scores for you and your best friend over the weekend. You could write a messy list on a scrap of paper, or you could use a super-organized math grid! 🎮
That grid is called a matrix (the plural is matrices). It is basically a magical mathematical box that holds numbers in rows (going across) and columns (going up and down). On the ISEE Upper Level, you might see these number grids and be asked to add them, subtract them, or multiply them by a single number.
Adding matrices is just like combining two orders of pizza. 🍕 If Matrix A is what you ate on Saturday (2 slices of pepperoni, 1 cheese) and Matrix B is what you ate on Sunday (3 pepperoni, 2 cheese), you just add the matching spots together! The number in the top-left corner of the first box adds to the top-left corner of the second box. The bottom-right adds to the bottom-right. It is just basic addition, but organized in a cool box!
There is one huge rule to remember: to add or subtract these math boxes, they must be the exact same size. A 2-by-2 box can only be added to another 2-by-2 box. If they do not match, you cannot combine them—just like you cannot play a two-player game with three controllers! 🕹️
Practice Questions
3 practice questions for ISEE Upper Level
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- To subtract matrices, subtract the corresponding elements in the same positions.
Top-left:
Top-right:
Bottom-left:
Bottom-right:
The resulting matrix is .
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- To multiply a matrix by a scalar (a single number outside the matrix), multiply every element inside the matrix by that scalar.
Top-left:
Top-right:
Bottom-left:
Bottom-right:
The resulting matrix is .
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- According to the order of operations, first perform the scalar multiplication on the first matrix: . Next, add this result to the second matrix by adding the corresponding elements.
Top-left:
Top-right:
Bottom-left:
Bottom-right:
The resulting matrix is .
Tips & Strategies
- Always check the dimensions first by reading Rows then Columns (think 'RC Cola'). If a question asks you to add a 2x2 matrix to a 2x3 matrix, the answer is 'cannot be determined' or 'undefined'!
- When multiplying a matrix by a negative number, draw a little star on your paper to remind yourself to flip the sign of EVERY number inside the matrix, not just the first one you see.
Common Mistakes
- Watch out for mixing up rows and columns! Remember, rows go left-to-right (like rows in a movie theater), and columns go up-and-down (like columns holding up a building).
- Don't forget that subtraction order matters! Just like is different from , Matrix A minus Matrix B is different from Matrix B minus Matrix A.
Frequently Asked Questions
Do I need to know how to multiply two matrices together for the ISEE?
Nope! For the ISEE Upper Level, you only need to know how to add them, subtract them, and multiply them by a single outside number. Multiplying a full matrix by another full matrix is saved for high school math!
What if I see a matrix question but I forget how to do it?
There is no penalty for guessing on the ISEE! If you are totally stuck, try to eliminate any answer choices that are the wrong size. Then, pick your favorite letter from the remaining choices (A, B, C, or D) and move on.
Are matrices on the Lower or Middle Level ISEE?
No, you do not need to worry about them! Matrices only pop up on the Upper Level ISEE, which is for students applying to grades 9 through 12.
Can fractions be inside a matrix?
Yes! A matrix is just a box that holds numbers, so it can hold whole numbers, negatives, and even fractions like . Just use your normal fraction rules when adding or subtracting the matching spots!