Add, Subtract and Scalar Multiply Matrices

Operations on Matrices such as adding, subtracting and multiplying by a scalar along with the equality of matrices are presented using examples and questions with solutions.

Page Content

• Add and Subtract Matrices
• Multiply a Matrix by a Scalar
• Equality of two Matrices
• Questions on Matrices
• Solutions to the Questions
• References

Add and Subtract Matrices

Only matrices of the same order can be added or subtracted. We add (or subtract) two matrices by adding (or subtracting) their corresponding entries.

Example 1

Solution

Multiply a Matrix by a Scalar

To multiply a matrix by a scalar, you multiply all entries of that matrix by the scalar. Note that scalar multiplication does not change the order of the matrix.
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Example 2

Solution

Equality of two Matrices

Two matrices are equal if they have the same order (equal number of rows and equal number of columns) and their corresponding entries are equal.

Example 3

a)   \( \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix} \), b)   \( \begin{bmatrix} x & 2x \\ 2 & 1 \\ \end{bmatrix} \), c)   \( \begin{bmatrix} 1 & -x \\ 3x & 5 \\ \end{bmatrix} \), d)   \( \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix} \), e)   \( \begin{bmatrix} 1 & 2 & 5\\ 3 & 4 & 8\\ \end{bmatrix} \),

Solution

Matrices a), b), c) and d) have the same order 2 by 2. But only matrices a) and d) are equal as they have equal corresponding entries. Matrix e) has the order 2 by 3 which is different to the order of all other matrices and is therefore not equal to any of the matrices.

Example 4

Solution

Questions on Matrices

• Part 1

  
  Find x and y, if possible, such that
  
  A) \( \begin{bmatrix} 2x + y & 3 & 10 \\ y + 1 & -2 & 0 \\ \end{bmatrix} = \begin{bmatrix} 2 & 3 & 10 \\ 3 & -2 & 0 \\ \end{bmatrix} \)
  
  B) \( \begin{bmatrix} 6 & -4 & -6 & x - y \end{bmatrix} = - 2 \begin{bmatrix} -3 & 2 & 2x + 2y & 13 \end{bmatrix} \)

• Part 2

  
  Write as one matrix
  
  A) \( \begin{bmatrix} -2 & 4 & 3 \\ -6 & 3 & -4 \\ -1 & 0 & 9 \\ \end{bmatrix} + \begin{bmatrix} -1 & 0 & 23 \\ -4 & -3 & 9 \\ 2 & -5 & 0 \\ \end{bmatrix} \)
  
  B) \( - 2 \begin{bmatrix} -2 & 3 \\ 1 & -5 \end{bmatrix} + 4 \begin{bmatrix} 3 & 4 \\ -2 & 9 \end{bmatrix} \)
  
  C) \( -3 \begin{bmatrix} -2x & y + 2 & 6\\ 4 & - x & -4 \\ \end{bmatrix} + 5 \begin{bmatrix} 3x & 5y & -5\\ 3y & x+y & -1 \end{bmatrix} \)

Solutions to the Questions

• Part 1

  
  A) The two matrices have the same order and to be equal, they need to have equal corresponding entries. Hence the simultaneous equations. 2x + y = 2 and y + 1 = 3
  Solve the above equations to obtain: x = 0 and y = 2.
  
  B) Multiply the second equation by -2 and rewrite the given equation as follows: \( \begin{bmatrix} 6 & -4 & -6 & x - y \end{bmatrix} = \begin{bmatrix} 6 & -4 & -2(2x + 2y) & -26 \end{bmatrix}\)
  For the two equations to be equal, we need to have
  -2(2x + 2y) = - 6 and x - y = - 26
  Solve the above system to obtain: x = -49/4 and y = 55/4

• Part 2

  
  A) \( \begin{bmatrix} -2 & 4 & 3 \\ -6 & 3 & -4 \\ -1 & 0 & 9 \end{bmatrix} + \begin{bmatrix} -1 & 0 & 23 \\ -4 & -3 & 9 \\ 2 & -5 & 0 \end{bmatrix} = \begin{bmatrix}-3&4&26\\ -10&0&5\\ 1&-5&9\end{bmatrix} \)
  
  B) \( - 2 \begin{bmatrix} -2 & 3 \\ 1 & -5 \end{bmatrix} + 4 \begin{bmatrix} 3 & 4 \\ -2 & 9 \end{bmatrix} = \begin{bmatrix}16&10\\ -10&46\end{bmatrix}\)
  
  C) \( -3 \begin{bmatrix} -2x & y + 2 & 6\\ 4 & - x & -4 \end{bmatrix} + 5 \begin{bmatrix} 3x & 5y & -5\\ 3y & x+y & -1 \end{bmatrix} = \begin{bmatrix}21x&22y-6&-43\\ -12+15y&8x+5y&7\end{bmatrix}\)

More References and links

• Matrices with Examples and Questions with Solutions
• Linear Algebra
• Matrix (mathematics)
• Matrices Applied to Electric Circuits
• Inverse Matrix Questions with Solutions