Studdy AI Logo
Math

QuestionFind the product matrix BCB C for matrices B=(−2305)B=\begin{pmatrix}-2 & 3 \\ 0 & 5\end{pmatrix} and C=(04−1301)C=\begin{pmatrix}0 & 4 & -1 \\ 3 & 0 & 1\end{pmatrix}.

Studdy Solution

STEP 1

Assumptions1. Matrix B is ax matrix with elements -,3,0,5. Matrix C is ax3 matrix with elements0,4, -1,3,0,13. We are required to find the product of matrices B and C, denoted as BC

STEP 2

In order to multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. In this case, matrix B has2 columns and matrix C has2 rows, so we can multiply them.

STEP 3

The product of two matrices is obtained by taking the dot product of the rows of the first matrix with the columns of the second matrix. The (i, j) element of the product matrix is the dot product of the i-th row of the first matrix and the j-th column of the second matrix.

STEP 4

Let's start by finding the first element of the product matrix BC. This is the dot product of the first row of B and the first column of C.
BC11=B11C11+B12C21BC_{11} = B_{11}C_{11} + B_{12}C_{21}

STEP 5

Substitute the values of the elements into the equation.
BC11=−2∗0+3∗3BC_{11} = -2*0 +3*3

STEP 6

Calculate the value of BC_{11}.
BC11=0+9=9BC_{11} =0 +9 =9

STEP 7

Let's find the second element of the first row of the product matrix BC. This is the dot product of the first row of B and the second column of C.
BC12=B11C12+B12C22BC_{12} = B_{11}C_{12} + B_{12}C_{22}

STEP 8

Substitute the values of the elements into the equation.
BC12=−2∗4+3∗0BC_{12} = -2*4 +3*0

STEP 9

Calculate the value of BC_{12}.
BC12=−8+=−8BC_{12} = -8 + = -8

STEP 10

Let's find the third element of the first row of the product matrix BC. This is the dot product of the first row of B and the third column of C.
BC13=BC13+B12C23BC_{13} = B_{}C_{13} + B_{12}C_{23}

STEP 11

Substitute the values of the elements into the equation.
BC13=−∗(−)+3∗BC_{13} = -*(-) +3*

STEP 12

Calculate the value of BC_{}.
BC=2+=5BC_{} =2 + =5

STEP 13

Let's find the first element of the second row of the product matrix BC. This is the dot product of the second row of B and the first column of C.
BC21=B21C11+B22C21BC_{21} = B_{21}C_{11} + B_{22}C_{21}

STEP 14

Substitute the values of the elements into the equation.
BC21=0∗0+∗3BC_{21} =0*0 +*3

STEP 15

Calculate the value of BC_{21}.
BC21=0+15=15BC_{21} =0 +15 =15

STEP 16

Let's find the second element of the second row of the product matrix BC. This is the dot product of the second row of B and the second column of C.
BC22=B21C12+B22C22BC_{22} = B_{21}C_{12} + B_{22}C_{22}

STEP 17

Substitute the values of the elements into the equation.
BC22=0∗4+5∗0BC_{22} =0*4 +5*0

STEP 18

Calculate the value of BC_{22}.
BC22=0+0=0BC_{22} =0 +0 =0

STEP 19

Let's find the third element of the second row of the product matrix BC. This is the dot product of the second row of B and the third column of C.
BC23=B21C13+B22C23BC_{23} = B_{21}C_{13} + B_{22}C_{23}

STEP 20

Substitute the values of the elements into the equation.
BC23=0∗(−)+5∗BC_{23} =0*(-) +5*

STEP 21

Calculate the value of BC_{23}.
BC23=0+5=5BC_{23} =0 +5 =5

STEP 22

Now that we have all the elements of the product matrix BC, we can write it out.
BC=[BC11BC12BC13BC21BC22BC]BC = \left[\begin{array}{ccc} BC_{11} & BC_{12} & BC_{13} \\ BC_{21} & BC_{22} & BC_{} \end{array}\right]

STEP 23

Substitute the calculated values into the matrix.
BC=[9−851505]BC = \left[\begin{array}{ccc} 9 & -8 &5 \\ 15 &0 &5\end{array}\right]The product matrix BC isBC=[9−851505]BC = \left[\begin{array}{ccc} 9 & -8 &5 \\ 15 &0 &5\end{array}\right]

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord