Questions: Calculate the scalar multiplication of matrix A: Step 1 Now to find 2A-3B, first find 2A and then 3B using the definition of the product of a real number and a matrix. 2A=2 0 -1 3 -1 -2 -1 = -2 -2 3B=3 -3 1 2 2 5 -3 = 3 6 6: 1

Calculate the scalar multiplication of matrix A:
Step 1

Now to find 2A-3B, first find 2A and then 3B using the definition of the product of a real number and a matrix.

2A=2
0 -1 3
-1 -2 -1

=
-2
-2

3B=3
-3 1 2
2 5 -3

=
3
6
6:
1

Transcript text: Calculate the scalar multiplication of matrix A: Step 1 Now to find $2 A-3 B$, first find $2 A$ and then $3 B$ using the definition of the product of a real number and a matrix. \[ \begin{array}{l} 2 A=2\left[\begin{array}{ccc} 0 & -1 & 3 \\ -1 & -2 & -1 \end{array}\right] \\ =\left[\begin{array}{lll} \square & -2 & \square \\ -2 & \square & \square \end{array}\right] \\ 3 B=3\left[\begin{array}{rrc} -3 & 1 & 2 \\ 2 & 5 & -3 \end{array}\right] \\ =\left[\begin{array}{lc} \square & 3 \\ 6 & \square \end{array}\right. \\ \left.\begin{array}{lll} 6: & \\ \hdashline & & 1 \end{array}\right] \end{array} \]

Solution

Solution Steps

To solve the problem of finding $2A - 3B$, we first perform scalar multiplication on matrices $A$ and $B$. Multiply each element of matrix $A$ by 2 to get $2A$, and each element of matrix $B$ by 3 to get $3B$. Finally, subtract the resulting matrix $3B$ from $2A$.

Step 1: Scalar Multiplication of Matrix $A$

To find $2A$, multiply each element of matrix $A$ by 2:

\[ A = \begin{bmatrix} 0 & -1 & 3 \\ -1 & -2 & -1 \end{bmatrix} \]

\[ 2A = 2 \times \begin{bmatrix} 0 & -1 & 3 \\ -1 & -2 & -1 \end{bmatrix} = \begin{bmatrix} 0 & -2 & 6 \\ -2 & -4 & -2 \end{bmatrix} \]

Step 2: Scalar Multiplication of Matrix $B$

To find $3B$, multiply each element of matrix $B$ by 3:

\[ B = \begin{bmatrix} -3 & 1 & 2 \\ 2 & 5 & -3 \end{bmatrix} \]

\[ 3B = 3 \times \begin{bmatrix} -3 & 1 & 2 \\ 2 & 5 & -3 \end{bmatrix} = \begin{bmatrix} -9 & 3 & 6 \\ 6 & 15 & -9 \end{bmatrix} \]

Step 3: Subtract Matrices

Subtract matrix $3B$ from matrix $2A$:

\[ 2A - 3B = \begin{bmatrix} 0 & -2 & 6 \\ -2 & -4 & -2 \end{bmatrix} - \begin{bmatrix} -9 & 3 & 6 \\ 6 & 15 & -9 \end{bmatrix} \]

\[ = \begin{bmatrix} 0 - (-9) & -2 - 3 & 6 - 6 \\ -2 - 6 & -4 - 15 & -2 - (-9) \end{bmatrix} \]