ACT Math : Scalar interactions with Matrices

Example Questions

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Example Question #1 : Matrices

Evaluate:

Explanation:

This problem involves a scalar multiplication with a matrix. Simply distribute the negative three and multiply this value with every number in the 2 by 3 matrix. The rows and columns will not change.

Example Question #2 : Matrices

What is ?

Explanation:

You can begin by treating this equation just like it was:

That is, you can divide both sides by :

Now, for scalar multiplication of matrices, you merely need to multiply the scalar by each component:

Then, simplify:

Therefore,

Example Question #3 : Matrices

If , what is ?

Explanation:

Begin by distributing the fraction through the matrix on the left side of the equation. This will simplify the contents, given that they are factors of :

Now, this means that your equation looks like:

This simply means:

and

or

Therefore,

Example Question #4 : Matrices

Simplify:

Explanation:

Scalar multiplication and addition of matrices are both very easy. Just like regular scalar values, you do multiplication first:

The addition of matrices is very easy. You merely need to add them directly together, correlating the spaces directly.

Example Question #5 : Matrices

Simplify the following

Explanation:

When multplying any matrix by a scalar quantity (3 in our case), we simply multiply each term in the matrix by the scalar.

Therefore, every number simply gets multiplied by 3, giving us our answer.

Example Question #6 : Matrices

Define matrix , and let  be the 3x3 identity matrix.

If , then evaluate .

Explanation:

The 3x3 identity matrix is

Both scalar multplication of a matrix and matrix addition are performed elementwise, so

is the first element in the third row of , which is 3; similarly, . Therefore,

Example Question #7 : Matrices

Define matrix , and let  be the 3x3 identity matrix.

If , then evaluate .

Explanation:

The 3x3 identity matrix is

Both scalar multplication of a matrix and matrix addition are performed elementwise, so

is the first element in the third row of , which is 3; similarly, . Therefore,

Example Question #8 : Matrices

Define matrix .

If , evaluate  .

The correct answer is not among the other responses.

Explanation:

If , then .

Scalar multplication of a matrix is done elementwise, so

is the first element in the second row of , which is 5, so

Example Question #9 : Matrices

Define matrix .

If , evaluate  .

The correct answer is not among the other responses.

Explanation:

Scalar multplication of a matrix is done elementwise, so

is the third element in the second row of , which is 1, so

Example Question #10 : Matrices

Define matrix , and let  be the 3x3 identity matrix.

If , evaluate .

The correct answer is not given among the other responses.