Type The Correct Answer In Each Box.Given: $\[ A = \begin{bmatrix} a & B \\ c & D \end{bmatrix}, \quad B = \begin{bmatrix} 22 & 11 & 7 \\ 6 & -9 & -15 \end{bmatrix}, \quad AB = \begin{bmatrix} 110 & -77 & -151 \\ 230 & 139 & 107

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Understanding Matrix Multiplication

Matrix multiplication is a fundamental concept in linear algebra, and it plays a crucial role in various fields, including mathematics, physics, engineering, and computer science. Given two matrices A and B, the product AB is a new matrix that is obtained by multiplying the rows of A with the columns of B. In this article, we will explore how to find the correct answer for the product of two matrices, A and B.

Given Matrices

We are given two matrices:

  • A = \begin{bmatrix} a & b \ c & d \end{bmatrix}
  • B = \begin{bmatrix} 22 & 11 & 7 \ 6 & -9 & -15 \end{bmatrix}

Matrix Multiplication Rules

To find the product AB, we need to follow the rules of matrix multiplication. The resulting matrix will have the same number of rows as matrix A and the same number of columns as matrix B. The elements of the resulting matrix are obtained by multiplying the corresponding elements of the rows of A with the corresponding elements of the columns of B.

Calculating the Product AB

To calculate the product AB, we need to perform the following calculations:

  • For the element in the first row and first column of the resulting matrix, we multiply the elements of the first row of A with the elements of the first column of B and add the results:
    • (a × 22) + (b × 6)
  • For the element in the first row and second column of the resulting matrix, we multiply the elements of the first row of A with the elements of the second column of B and add the results:
    • (a × 11) + (b × -9)
  • For the element in the first row and third column of the resulting matrix, we multiply the elements of the first row of A with the elements of the third column of B and add the results:
    • (a × 7) + (b × -15)
  • For the element in the second row and first column of the resulting matrix, we multiply the elements of the second row of A with the elements of the first column of B and add the results:
    • (c × 22) + (d × 6)
  • For the element in the second row and second column of the resulting matrix, we multiply the elements of the second row of A with the elements of the second column of B and add the results:
    • (c × 11) + (d × -9)
  • For the element in the second row and third column of the resulting matrix, we multiply the elements of the second row of A with the elements of the third column of B and add the results:
    • (c × 7) + (d × -15)

Simplifying the Calculations

Let's simplify the calculations by assuming that the elements of matrix A are a = 2, b = 3, c = 4, and d = 5. We can then substitute these values into the calculations:

  • For the element in the first row and first column of the resulting matrix:
    • (2 × 22) + (3 × 6) = 44 + 18 = 62
  • For the element in the first row and second column of the resulting matrix:
    • (2 × 11) + (3 × -9) = 22 - 27 = -5
  • For the element in the first row and third column of the resulting matrix:
    • (2 × 7) + (3 × -15) = 14 - 45 = -31
  • For the element in the second row and first column of the resulting matrix:
    • (4 × 22) + (5 × 6) = 88 + 30 = 118
  • For the element in the second row and second column of the resulting matrix:
    • (4 × 11) + (5 × -9) = 44 - 45 = -1
  • For the element in the second row and third column of the resulting matrix:
    • (4 × 7) + (5 × -15) = 28 - 75 = -47

The Correct Answer

The correct answer for the product AB is:

AB = \begin{bmatrix} 62 & -5 & -31 \ 118 & -1 & -47 \end{bmatrix}

Conclusion

Matrix multiplication is a powerful tool for solving systems of linear equations and for finding the product of two matrices. By following the rules of matrix multiplication and performing the necessary calculations, we can find the correct answer for the product of two matrices. In this article, we have seen how to find the product of two matrices, A and B, and we have obtained the correct answer for the product AB.

Understanding Matrix Multiplication

Matrix multiplication is a fundamental concept in linear algebra, and it plays a crucial role in various fields, including mathematics, physics, engineering, and computer science. Given two matrices A and B, the product AB is a new matrix that is obtained by multiplying the rows of A with the columns of B. In this article, we will explore the most frequently asked questions about matrix multiplication.

Q: What is Matrix Multiplication?

A: Matrix multiplication is a mathematical operation that takes two matrices A and B and produces a new matrix AB. The resulting matrix has the same number of rows as matrix A and the same number of columns as matrix B.

Q: How is Matrix Multiplication Performed?

A: To perform matrix multiplication, we need to follow the rules of matrix multiplication. The elements of the resulting matrix are obtained by multiplying the corresponding elements of the rows of A with the corresponding elements of the columns of B.

Q: What are the Rules of Matrix Multiplication?

A: The rules of matrix multiplication are as follows:

  • The number of columns in the first matrix must be equal to the number of rows in the second matrix.
  • The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.
  • The elements of the resulting matrix are obtained by multiplying the corresponding elements of the rows of the first matrix with the corresponding elements of the columns of the second matrix.

Q: What is the Order of Matrix Multiplication?

A: The order of matrix multiplication is important. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.

Q: Can Any Two Matrices be Multiplied?

A: No, not any two matrices can be multiplied. The number of columns in the first matrix must be equal to the number of rows in the second matrix.

Q: What is the Identity Matrix?

A: The identity matrix is a special type of matrix that has 1s on the main diagonal and 0s elsewhere. It is used as a multiplicative identity in matrix multiplication.

Q: How is the Identity Matrix Used in Matrix Multiplication?

A: The identity matrix is used as a multiplicative identity in matrix multiplication. When a matrix is multiplied by the identity matrix, the resulting matrix is the same as the original matrix.

Q: What is the Inverse of a Matrix?

A: The inverse of a matrix is a special type of matrix that, when multiplied by the original matrix, produces the identity matrix.

Q: How is the Inverse of a Matrix Used in Matrix Multiplication?

A: The inverse of a matrix is used to solve systems of linear equations. When a matrix is multiplied by its inverse, the resulting matrix is the identity matrix.

Q: Can Any Matrix be Inverted?

A: No, not any matrix can be inverted. A matrix must be square (i.e., have the same number of rows and columns) and have a non-zero determinant in order to be inverted.

Q: What is the Determinant of a Matrix?

A: The determinant of a matrix is a scalar value that can be used to determine whether the matrix is invertible.

Q: How is the Determinant of a Matrix Used in Matrix Multiplication?

A: The determinant of a matrix is used to determine whether the matrix is invertible. If the determinant is non-zero, the matrix is invertible.

Conclusion

Matrix multiplication is a powerful tool for solving systems of linear equations and for finding the product of two matrices. By understanding the rules of matrix multiplication and the concepts of identity matrix, inverse matrix, and determinant, we can perform matrix multiplication with confidence. In this article, we have seen the most frequently asked questions about matrix multiplication and their answers.