Matrices and Matrix Operations

21 questions

Matrices $C$ and $D$ are defined as follows:

C = \begin{bmatrix} 4 & 1 \\ -2 & k \end{bmatrix}, \quad D = \begin{bmatrix} 3 & -1 \\ 5 & 2 \end{bmatrix}

If the element in the second row and first column of the product matrix $CD$ is $-11$ , what is the value of $k$ ?

Show answer & explanation

Answer: -1

The value of

k

-1

To find the element in the second row and first column of the product matrix

CD

, we take the dot product of the second row of

C

, which is

\begin{bmatrix} -2 & k \end{bmatrix}

, and the first column of

D

, which is

\begin{bmatrix} 3 \\ 5 \end{bmatrix}

. This gives

(-2)(3) + (k)(5) = -6 + 5k

. Setting this equal to the given value of

-11

yields the equation

-6 + 5k = -11

. Adding

6

to both sides gives

5k = -5

, and dividing by

5

results in

k = -1

Identify the second row of matrix

C

and the first column of matrix

D

Row 2 of

C

\begin{bmatrix} -2 & k \end{bmatrix}

and Column 1 of

D

\begin{bmatrix} 3 \\ 5 \end{bmatrix}

To find the element in the second row and first column of the product matrix

CD

, we must compute the dot product of the second row of the first matrix (

C

) and the first column of the second matrix (

D

Multiply the corresponding elements of the row and column and add the products.

The element at row 2, column 1 of

CD

(-2)(3) + (k)(5) = -6 + 5k

This defines the matrix multiplication rule for that specific position in the resulting matrix.

Set the expression

-6 + 5k

equal to the given value of

-11

and solve the linear equation for

k

-6 + 5k = -11 \implies 5k = -5 \implies k = -1

Solving the equation yields the value of the unknown variable

k

2x2 Matrix Multiplication

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