What is true about [ -3 0 0 -3 ][ 1 -2 4 1 -1 2 ] and -3[ 1 -2

step 1 Let me move this up. So we're given this question over here, and we're asked to find which of, w

What is true about [ -3 0 0 -3 ][ 1 -2 ...

What is true about $$\left[\begin{array}{rr}{-3} & {0} \\ {0} & {-3}\end{array}\right]\left[\begin{array}{rrr}{1} & {-2} & {4} \\ {1} & {-1} & {2}\end{array}\right] and -3\left[\begin{array}{rrr}{1} & {-2} & {4} \\ {1} & {-1} & {2}\end{array}\right]$$ $$\begin{array}{l}{\text { A They are equal matrices. }} \\ {\text { C Both are matrices for reflection. }}\end{array}$$ $$ \begin{array}{l}{\text { B They are opposite matrices. }} \\ {\text { D Neither product exists. }}\end{array} $$

What is true about $$\left[\begin{array}{rr}{-3} & {0} \\ {0} & {-3}\end{array}\right]\left[\begin{array}{rrr}{1} & {-2} & {4} \\ {1} & {-1} & {2}\end{array}\right] and -3\left[\begin{array}{rrr}{1} & {-2} & {4} \\ {1} & {-1} & {2}\end{array}\right]$$

$$\begin{array}{l}{\text { A They are equal matrices. }} \\ {\text { C Both are matrices for reflection. }}\end{array}$$
$$
\begin{array}{l}{\text { B They are opposite matrices. }} \\ {\text { D Neither product exists. }}\end{array}
$$

Key Concepts

Matrix Multiplication Validity

Matrix multiplication is defined when the number of columns in the first matrix equals the number of rows in the second matrix. In the given problem, the dimensions of the matrices involved allow the product to exist, ensuring that the operation is properly defined. This is an important consideration when determining whether an operation involving matrices is valid.

Multiplication by a Diagonal Matrix

A diagonal matrix has nonzero elements only on its main diagonal. When a diagonal matrix with equal diagonal entries (i.e., a scalar times the identity matrix) multiplies another matrix, the effect is equivalent to multiplying every row of the second matrix by that scalar. This equivalence underpins the property that such a matrix product is the same as scalar multiplication of the given matrix.

Scalar Multiplication of Matrices

Scalar multiplication involves multiplying every entry of a matrix by the same scalar value. When a scalar multiplies a matrix, the resulting matrix is formed by taking each entry of the original matrix and multiplying it by that scalar. This property makes it straightforward to compare expressions where matrices are multiplied by scalars either externally or as a result of multiplying by a diagonal matrix that is a scalar multiple of the identity matrix.

Transcript

00:01 Let me move this up.

00:04 So we're given this question over here, and we're asked to find which of, we have to find, we have to find what is true.

00:15 So let's go through each of the options.

00:17 First option is they are equal matrices.

00:22 We can just test that out.

00:23 We can do the multiplication over here, and we can do the scalar multiplication here and see if they're equal.

00:30 And so we do that over here.

00:32 We multiply these two matrices, and i did the work already, but i guess i can go over it.

00:40 To find this element over here, we're going to do, well, first, let's select this.

00:45 That's a negative 3, right? so that's going to be a negative 3 times 1 plus 0 times 1, which is 0.

00:53 So it's going to be equal to negative 3.

00:56 Then we do it for this.

00:58 To obtain this one, to obtain this one, we're going to do negative.

01:02 3 times negative 2, 6 plus 0 times negative 1, which is 0, and so the whole thing is equal to 6.

01:12 Lastly, we'll do this again.

01:15 We'll go negative 3 times 4, which is negative, it's negative 12 plus 0 times 2, which is 0 is equal to negative 12.

01:25 So same thing over here.

01:27 We want to find second row first column.

01:33 So we're going to select the second row here, the first column here, and we're going to multiply the corresponding elements and add them.

01:45 So 0 times 1 is 1 plus negative, or, yeah, 0 times 1 is 0 plus negative 1 is 0 plus negative.

01:57 3.

01:57 So that's a negative 3.

01:59 0 times negative 2 is 0 plus negative 3 times negative 1 is 3.

02:03 And 0 times 4 is 0 plus negative 3 times 2, which is negative 6.

02:09 So this is where we get over here.

02:13 Wait, let me check that one again.

02:15 0 times 4 is plus 0 plus 0 plus 0 plus negative 3 times 2, which is negative 6 is equal to negative 6.

02:30 So we get that.

02:33 And let's do scalar multiplication over here.

02:37 And to do scalar multiplication, you basically multiply each element in the matrix by whatever is over here.

02:43 So one time, to get the corresponding element.

02:46 So to get the corresponding element here, we're just going to multiply this element over here by the factor, scale or factor over here.

02:54 So that's going to be negative 3.

02:56 This is going to be negative 2, negative 2 times negative 3 will give us...

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